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Exercises 31 and 32 concern finite-dimensional vector spaces $V$ and $W$ and a linear transformation $T : V \rightarrow W$.Let $H$ be a nonzero subspace of $V,$ and let $T(H)$ be the set of images of vectors in $H .$ Then $T(H)$ is a subspace of $W,$ by Exercise 35 in Section $4.2 .$ Prove that $\operatorname{dim} T(H) \leq \operatorname{dim} H .$

see the proof

Calculus 3

Chapter 4

Vector Spaces

Section 5

The Dimension of a Vector Space

Vectors

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Lectures

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In mathematics, a vector (…

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Exercises 31 and 32 concer…

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Exercises 31 and 32 reveal…

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Let $V$ and $W$ be vector …

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Let $H$ be an $n$ -dimensi…

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Given $T : V \rightarrow W…

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Let $H$ be the set of all …

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Let $W$ be the set of all …

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In Exercises 29 and $30, V…

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Exercises $23-26$ concern …

So this problem we're given transfer in your transformation from V. Too tough for you. Where B and W are finite dimensional. Just say fine. Nine and h subspace off B and age. It's not zero and t is 121 Okay, so we want to show that I want to show you mention team age is equal to dimension off. H all right. So, uh, one thing to notice that if we have have a basis, um, say is itself a JJ. Let's say so. Be one thio the end. So we are considering the mapping off these vectors. Then you won and be, too until p t are also we nearly independent the reason the reason for why this is still lingering. And then it's because, uh, tea is ah, is that 1 to 1? Maybe so. Note that tea is 121 So it turns out this these vectors, the mapping off we want to be p will spend. It's been tee off me, uh, d of H. And since they are p vectors here, so the number of factors is so you mentioned off age. It's equal to and shin of t age, which I won't. Hey, so we're done now, given that addition no additional condition. Say he is on two. So what do we have now? Yeah. Uh, uh. T zone two can directly take, um TV to be w so by our previous, like our previous, uh um, permission. We know that, um, you mention, uh, TV is dimension off W Well, actually, actually, this is a dimension of TV should be equal to dimension. Enough T and dimension of TV is actually dimension enough. W There we go. Curious our our approach to prove that commission of ws dimension is secretive. Dimension off be if if we have tees off.

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