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  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 $\mathit P_6(\Bbb R)$

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  • $\begingroup$ What do you mean by $\times_3$? I'm not familiar with that notation. $\endgroup$ Commented May 12, 2020 at 19:05
  • $\begingroup$ sorry,that is 3x3 matrix $\endgroup$
    – user781604
    Commented May 12, 2020 at 19:06
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    $\begingroup$ no, probably polynomials of degree at most k. projective space seems unlikely to appear in such a question $\endgroup$ Commented May 12, 2020 at 19:12
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    $\begingroup$ @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. $\endgroup$ Commented May 12, 2020 at 19:17
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    $\begingroup$ 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 $n.$ $\endgroup$ Commented May 12, 2020 at 19:19

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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 $k$.

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  • $\begingroup$ Got it, thanks! $\endgroup$
    – user781604
    Commented May 12, 2020 at 19:23
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    $\begingroup$ Be careful with question 2: Nothing in it says that $A$ and $B$ must be square matrices. $\endgroup$
    – amd
    Commented May 12, 2020 at 20:36
  • $\begingroup$ 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. $\endgroup$
    – Isaac Ren
    Commented May 13, 2020 at 7:22

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