Linear Algebra - Matrix Transformations

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<div id="description" class="style-scope ytd-video-secondary-info-renderer">Matrix multiplication and linear algebra explained with 3D animations.</div>

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<h4 id="title" class="style-scope ytd-metadata-row-renderer">Caption author (French)</h4>

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<div id="content" class="style-scope ytd-metadata-row-renderer"><a class="yt-simple-endpoint style-scope yt-formatted-string" spellcheck="false" href="https://www.youtube.com/channel/UClJVEaJgKAtD0fGeTHxqXsA">Ignacio Burgos</a><a class="yt-simple-endpoint style-scope yt-formatted-string" spellcheck="false" href="https://www.youtube.com/channel/UCIGqFrNC6vIKKD_pKO2o3Ow">Eyder Aranda</a></div>

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<div class="cue style-scope ytd-transcript-body-renderer">Suppose we have a number.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">By changing the value of this one number,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">we can represent any point in a one dimensional space.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Now suppose we have two numbers.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">By changing the values of these two numbers,</div>

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<div class="cue style-scope ytd-transcript-body-renderer active">we can represent any point in a two dimensional space.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Now suppose we have three numbers.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">We can now represent any point in a three dimensional space.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Similarly, if we have a set of four numbers,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">we can represent any point in a four dimensional space.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">If we have a set of five numbers,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">we can represent a point in five dimensional space.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">And if we have an infinite set of numbers,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">we can represent a point in a space with an infinite number of dimensions.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Regardless of how many dimensions we have,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">each set of numbers which represents a point in space</div>

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<div class="cue style-scope ytd-transcript-body-renderer">can be thought of as an arrow, which we will call a vector.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Each vector has a length, and a direction.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">If we multiply the vector by a number, the vector’s length will change,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">but its direction will stay the same.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">If we add two vectors together, the result is as shown.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Each vector can be thought of as the</div>

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<div class="cue style-scope ytd-transcript-body-renderer">sum of vectors pointed in each of the different dimensions.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">And each of the vectors pointed in each of the dimensions can be thought of as</div>

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<div class="cue style-scope ytd-transcript-body-renderer">a vector a length of one multiplied by a number.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">There are many transformation that can be applied to a vector.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">For example, the vector can be rotated in space around this axis,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">by a certain number of degrees.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Suppose we apply this rotation to a group of vectors.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">And then we add all the vectors together.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Now, suppose that instead of starting out by rotating the vectors,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">we instead first add all of all the vectors together,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">and then afterwards we rotate the result.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">It turns out that the final vector that is produced is the same as before.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">If the final vector that is produced is the same regardless of</div>

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<div class="cue style-scope ytd-transcript-body-renderer">whether the vectors are added together before or after the transformation,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">then we say that this is a linear transformation.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Another example of a linear transformation is having</div>

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<div class="cue style-scope ytd-transcript-body-renderer">the vector stretched or squeezed as shown.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Or having the vector flipped into this mirror image of itself.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Or having the vector stretched in this dimension,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">and then flipped.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">In each of these cases, the final vector that is produced is the same regardless</div>

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<div class="cue style-scope ytd-transcript-body-renderer">of whether the vectors are added together before or after the transformation.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">In each of these cases, the final vector that is produced is the same regardless</div>

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<div class="cue style-scope ytd-transcript-body-renderer">of whether the vectors are added together before or after the transformation.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">This is the definition of a linear transformation.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">We can represent a linear transformation mathematically by showing</div>

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<div class="cue style-scope ytd-transcript-body-renderer">how the transformation would affect vectors with a length of one,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">pointed in each of the dimensions.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">The original vectors are vectors with a length of one,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">pointed in each of the dimensions.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">After the transformation is applied, the new vectors are the result.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Suppose we take each of the new vectors that result from this,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">and we line up all their numbers together as is shown.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">We now have what we call a matrix.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">All linear transformations can be represented with a matrix.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">When we apply the transformation, we say that we are</div>

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<div class="cue style-scope ytd-transcript-body-renderer">multiplying the vector by the matrix, and we represent it like this.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">We defined the numbers in each column of the matrix based on the result of</div>

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<div class="cue style-scope ytd-transcript-body-renderer">the transformation of vectors of length one in each of the dimensions.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Therefore, when the matrix is multiplied by a vector of length one</div>

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<div class="cue style-scope ytd-transcript-body-renderer">in one of the dimensions, the result is a follows.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">If we want to find out how the transformation affects a vector pointed</div>

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<div class="cue style-scope ytd-transcript-body-renderer">in one of the dimensions that is not length one, then we can do the following.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">And if we want to find out how the transformation affects any vector</div>

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<div class="cue style-scope ytd-transcript-body-renderer">of any length pointed in any direction, then we can do this as shown.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">This is how a matrix is multiplied by a vector,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">and it is how all linear transformations can be computed.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">This is true for four dimensions</div>

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<div class="cue style-scope ytd-transcript-body-renderer">For Five dimensions</div>

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<div class="cue style-scope ytd-transcript-body-renderer">And for an infinite number of dimensions</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Much more information about mathematics</div>

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<div class="cue style-scope ytd-transcript-body-renderer">is available in the other videos on this channel.</div>

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Linear Algebra - Matrix Transformations

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