# Are these linear algebra questions true or false?

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?

• Is this a school assignment? – Jack Pfaffinger Apr 5 at 19:47
• No it's a quiz I took. I want to calculate my grade – user660922 Apr 5 at 19:48
• What is $\Bbb P_4$? – Bernard Apr 5 at 20:05
• The vector space of polynomials of degree less than or equal to $4$ – user660922 Apr 5 at 20:13

(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!