# True of False-linear algebra

1. If $$T: M_{3\times3} (\Bbb R)\rightarrow \mathit P_4(\Bbb R)$$ is a lnear map, and the nullity ($$T$$)=4, then $$T$$ is onto.

2. If the matrices A and B satisfy $$AB=I_n$$, then they are both invertible.

3. $$M_{2\times3} (\Bbb R)$$ is isomorphic to $$\mathit P_5(\Bbb R)$$

I think the first one is false. I know dim(rank)=5. Second one might be true; Third one might be false,it should be $$\mathit P_6(\Bbb R)$$

• What do you mean by $\times_3$? I'm not familiar with that notation. Commented May 12, 2020 at 19:05
• sorry,that is 3x3 matrix Commented May 12, 2020 at 19:06
• no, probably polynomials of degree at most k. projective space seems unlikely to appear in such a question Commented May 12, 2020 at 19:12
• @user781604 Issac was asking about what the notation $P_{k}(\mathbb{R})$ stood for. I said that it most likely is the space of polynomials with real coefficients such that the degree is at most k. Commented May 12, 2020 at 19:17
• The space $P_n(\mathbb{R})$ of polynomials of degree at most $n$ has dimension $n+1.$ A basis is $1, x, x^2, \ldots , x^n.$ If Rank ($T$) is 5, then the dimension of the image of $T$ is 5. How many subspaces of dimension 5 can a 5-dimensional space have? For the second question, $AB=I$ implies $BA=I,$ at least for finite $n.$ Commented May 12, 2020 at 19:19

As Nicholas said in the comments, I'll assume $$P_k(\Bbb R)$$ is the space of polynomials of degree at most $$k$$ (I've seen the notation $$\Bbb R_k[X]$$).

1. The rank of $$T$$ is $$5$$, as you say. $$P_4(\Bbb R)$$ is of dimension $$5$$, so $$T$$ is onto.
2. Calculate $$\require{enclose}\enclose{horizontalstrike}{\det(AB)=\det(A)\det(B)=1}$$, so $$A$$ and $$B$$ both have nonzero determinant, so are invertible. $$A$$ and $$B$$ are not necessarily square, and therefore are not necessarily invertible.
3. $$M_{2,3}(\Bbb R)$$ and $$P_5(\Bbb R)$$ are both real vector spaces of dimension $$6$$, so they are isomorphic.

One thing to remember is that the space of polynomials of degree up to $$k$$ is of dimension $$k+1$$, and not $$k$$.

• Got it, thanks! Commented May 12, 2020 at 19:23
• Be careful with question 2: Nothing in it says that $A$ and $B$ must be square matrices.
– amd
Commented May 12, 2020 at 20:36
• True. I thought about that at first, but decided that no one would talk about invertibility outside of (fully) multiplicative structures. I'll change my answer. Commented May 13, 2020 at 7:22