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Find the dimension of the subspace of all vectors in $\mathbb{R}^{3}$ whose first and third entries are equal.

2

Calculus 3

Chapter 4

Vector Spaces

Section 5

The Dimension of a Vector Space

Vectors

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In mathematics, a vector (…

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Let $S$ be the subspace of…

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determine a linearly indep…

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Show that the set of vecto…

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Give a geometric descripti…

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Hello. For this question, we need to find the dimension of a space which I've just arbitrarily called W, which is the set of all vectors in our three where the 1st and 3rd entry are equal. So if we look at this this Vek, this vector that I've written a b a all vectors which are in our space w could be written in this form. We can break it up into a times a vector plus b times a vector. On those vectors are 101 and 010 Which means that, uh, W is actually all of this implies that w is the span of two vectors 101 and 010 And this immediately implies that the dimension of W is to because this span of any set of vectors is a vector space whose dimension is equal to the number of linearly independent vectors and span. And clearly these two vectors air literally independent eso We know that the dimension of W is to another way to think about it in a less precise sense is that if the 1st and 3rd entries are equal, then I have two choices for my Constance because I can set the second slot, whatever I want. And I can set the first slot, whatever I want. But that forces the third slot to be, ah, specific value. So the dimension, the number of choices I have for Constance is said it too. But this is, ah, more formal way of expressing that idea. And it also gives you the two vectors, um, that represent this factor space and that.

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