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?
Best Answer
(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!