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Can you suggest me a collection of true-false linear algebra questions, like the ones found in the MIT exams, if possible with solutions (i.e. explanations)?

Sorry if it turns out that my request is off-topic, but if it is, please suggest where to ask it properly

A question I am fond of asking my students: Does the vector space of polynomials of degree $leq n$ have a basis consisting of polynomials with same degree?

Answer: yes.

Don't know how difficult these need to be. Here are some rather simple ones:

1. For every linear map $f : mathbb R^m to mathbb R^n$ there is a unique $mtimes n$ matrix with $f(v) = Av$ for all $vin mathbb R^m$.
   Answer: No, its actually an $ntimes m$ matrix. The expression $Av$ would not even make sense.


2. If $AB = AC$ and $Aneq 0$, then $B = C$ for all $ntimes n$ matrices $A,B,C$. Answer: No, take $n = 2$, $A = B = beginpmatrix 0 & 1 \ 0 & 0 endpmatrix$ and $C = 0$.


3. The determinant of a matrix with integer entries is an integer. Answer: Yes, take a look at (for example) the Leibniz formula for determinants.


4. 
   $det : mathbbR^ntimes n to mathbbR$ is a linear map for all $n$. Answer: No, unless $n = 1$. Instead we have $det(cA) = c^ndet(A)$ for $cin mathbb R$ and no "obvious rule" to compute $det(A+B)$.


5. We have $(A+B)^n = sum_i=1^n binomni A^n-i B^i$ for all $ntimes n$ matrices $A,B$. Answer: No. It is true, if $AB = BA$.


6. If $m,n geq 0, A in mathbb R ^mtimes n, Bin mathbb R^ ntimes m$ with $AB = I_m$ and $BA = I_n$ then $m=n$. Answer: Yes, this can be seen using the dimension formula / rank-nullity theorem.

A simple but not trivial question:

(T/F): If $ntimes n$ matrix $A$ is of rank $n$, then $A$ is diagonalizable by a similarity transformation $$ D = P^-1AP$$

Another, slightly harder one:

If $P(x)$ is a polynomial and $A$ is a matrix with eigenvalues $lambda_1 ldots lambda_n$, then $P(A)$ has eigenvalues $P(lambda_1) ldots P(lambda_n)$

If $T: mathbbR^3 to mathbbR$ is a linear transformation, then $T(x,y,z) = ax + by + cz$ for some $a,b,c in mathbbR$.

If $S,T: V to V$ are linear transformations and $v in ker T$, then $v in ker S circ T$.

If two non-zero vectors are linearly independent, then one must be a scalar multiple of the other.

If $T$ is invertible, then $ker T = textbf0$.

If $v_1, v_2, dots, v_n$ are linearly independent in $mathbbQ^n$, then they must form a basis for $mathbbQ^n$.

There can never be a linear map between finite dimensional spaces of different dimension.

Here are some random ones that have a narrower scope: they concern more about matrices than about linear algebra in general.

1. If a system of homogeneous linear equations is overdetermined (i.e. it has more equations than variables), it must have only the trivial solution. (False. Consider $Ax=0$, where $A$ has linearly dependent columns.)


2. If $A$ and $B$ are two matrices such that $AB=I$, then $BA=I$. (False. If $A,B$ are rectangular matrices and $AB=I$, then $BA$ must have deficient rank.)


3. Let $A,B$ be two rectangular matrices and $C=AB^top$. If $C^3-5C+I=0$, then $A$ has linearly independent rows. (True. $C$ is invertible and $A$ has full row rank.)


4. There exist four vectors $u,v,w,xinmathbb R^4$ such that $uv^top+vw^top+wx^top+xu^T=I_4$.
   (False. Let $P$ be the augmented matrix $(u,v,w,x)$ and
   $$
   C=pmatrix0&1&0&0\ 0&0&1&0\ 0&0&0&1\ 1&0&0&0.
   $$
   Then $uv^top+vw^top+wx^top+xu^T=PCP^T$ and its determinant is $-det(P)^2$ that cannot be 1.)


5. $vv^top$ is diagonalisable for any complex vector $v$. (False. Consider $v=pmatrix1\ i$. Then $vv^top$ is nonzero but nilpotent -- its square is zero, hence non-diagonalisable.)


6. If $P$ is an invertible matrix of the same size as $A$, then $operatornameadj(PAP^-1)=Poperatornameadj(A)P^-1$. (True, because $operatornameadj(M)$ is a polynomial in $M$.)


7. 
   

   Let $A,Bin M_3(mathbb R)$ and $ABA=A^2$. In each of the following cases, is it true that $operatornameadj(AB)=operatornameadj(A)B$?

   
   
   
   • $A$ is nonsingular.
   
   
   • $A$ has rank 2.
   
   
   • $A$ has rank 1.
   
   

   (True, false, true.) The first case is trivial because $B=I$. In the second case, consider the case where $A=operatornamediag(1,1,0)$ and $B-A$ is a matrix whose only nonzero row is the last one. In the third case, the rank deficiencies in both $A$ and $AB$ are at least 2. Hence their adjugate matrices are zero.

   
   


8. Let $varepsilon>0$ and $p$ be a polynomial with real coefficients such that $p(A)=0$ for every real square matrix $A$ that is entrywise smaller than $varepsilon$. Then $p=0$. (True. As $p(tI)=0$ when $t$ is small, we get $operatornamediag(p(t),ldots,p(t))=0$ when $t$ is small. Hence $p$ has infinitely many roots.)


9. Let $V$ be the vector space of all cubic polynomials with real coefficients. Let $T$ be the linear operator defined by $T(p)(x)=p'(x)$. Then its adjugate is $operatornameadj(T)=T^3$. (False. The characteristic polynomial of $T$ is $x^4$. Hence $operatornameadj(T)=-T^3$.)