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Which of the following are true or false?

(a) Let $H$ be a five-dimensional subspace of the ten-dimensional vector space $V$. Then, every set containing seven vectors from $H$ must be linearly dependent.

(b) Let $\mathcal{B}$ be a basis of $\mathbb{R}^{n}$. Let $A$ be a the matrix whose columns are the vectors in $\mathcal{B}$. Then, for every vector $x \in \mathbb{R}^{n}$, it is true that $[x]_{\mathcal{B}} = Ax$.

(c) The dimension of the vector space $\mathbb{P}_{4}$ is $4$.

(d) Let $A$ be a $3\times 3$ matrix, and let $H$ be the set of fixed vectors of $A$, that is, the set of $x \in \mathbb{R}^{3}$ for which $Ax = x$. Then $H$ is a subspace of $\mathbb{R}^{3}$.


I think (a) is false, but I don't really have a reason why. I think this is true because if you have an $n$-dimensional space, you only need $n$ linearly independent ones to span the space.

I think (b) is true since it looks like a definition I saw in my book. I don't have a reason why.

I know (c) is false. I'm pretty sure it's dimension 5. I have seen the proof before. EDIT: Proof here: Determining Bases of P4

I think (d) is true. Because if $Ax = x$ and $Ay = y$ then $A(x + y) = x + y$. Also the with scalar multiplication. Also, this is sort of like eigenvalues, from my understanding.

Can someone help me verify these please?

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    $\begingroup$ Is this a school assignment? $\endgroup$ – Jack Pfaffinger Apr 5 at 19:47
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    $\begingroup$ No it's a quiz I took. I want to calculate my grade $\endgroup$ – user660922 Apr 5 at 19:48
  • $\begingroup$ What is $\Bbb P_4$? $\endgroup$ – Bernard Apr 5 at 20:05
  • $\begingroup$ The vector space of polynomials of degree less than or equal to $4$ $\endgroup$ – user660922 Apr 5 at 20:13
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(a) is true. If a set of $7$ vectors in $H$ were independent, they are a basis for a subspace of $H$, so that the $5$-dimensional space $H$ has a subspace of dimension $7$, which is absurd. (b) is true; $A$ is called a change of basis matrix. (c) is false for the reason you state, and (d) is true for the reason you state. Good thinking!

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