Linear Algebra - Lecture 17 - Matrix Transformations

In this lecutre, we will discuss matrix transformations, which are functions that arise from multiplying a matrix by a vector. We will establish some vocabulary that we will be using when discussing these types of functions.

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2 minutes before one, as a matter of fact.

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Straightforward and Detailed

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Thank you for this. Much appreciated.

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I got lost at the row reduce part

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Helpful

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Outstanding explanation 👍

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[@jvc_1]

Thank you so much, please can you explain row reduction

[@HamblinMath]

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[@sandeepneemkar]

How can we geometrically interpret that its not in range of T? I mean how to visualize the vectors whose transformation falls out of range of T?

[@xoppa09]

the range of a transformation is a subset of the codomain

[@HamblinMath]

True

[@nealdyxthercentino5576]

From the example 2:00, why is it R3 to R2?

[@jensmalzer6344]

Because the matrix is 2 x 3, you go from R3 to R2. So basically for an m x n matrix, you go from Rn to Rm

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❣️

[@detree]

Wdym pivot in the last column

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[@dudius3743]

how was our row reduction different between b & c? wouldnt they have the same answer?

[@HamblinMath]

The two vectors are different. There's a -5 in the "b" vector and a +5 in the "c" vector.

[@dudius3743]

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🤩

[@ColyneOwino]

Good

[@elinanikolopoulou1769]

T(x) is the image or the transformation ?

[@HamblinMath]

T is the transformation. If x is a vector, then T(x) is the image of x. Sometimes we'll refer to the transformation as "T(x)" (where x is just a generic variable, not a specific vector). This is similar to when we refer to a function as f or f(x) interchangeably.

[@Ary2345]

In simple words for image when you solve T(X) = Ax the thing you get as a result is the image .

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[@confused6526]

I wonder why mathematicians decide to use many jargons which refer to the same shit? Domain -> Codomain... is the same as ... Image -> Range.... T refers to f which is a function.... Heck....

[@HamblinMath]

As explained in the video, domain, codomain, image, and range all refer to different things. It is important to use precise language so that our meaning is clear. Since transformations are a special kind of function, we often (but not always) use "T" since it is the first letter of the word "transformation."

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[@kingnehmna7810]

Great easy-to-understand lecture! Thank you!!!