Daily LinAlg T/F Questions

Posting daily true/false questions relating to undergraduate linear algebra. Please comment proofs/counterexamples. Link below for question and solution bank.

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@FightHaven

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Happy Pi Day.

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T/F: it is possible that A is an invertible matrix and A is the standard matrix for a non-injective function

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Hard mode:
T/F: suppose A and B are matrices such that AB is defined. Any vector in col(AB) is also in col(A)?

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T/F: a vector may be in colA and also in Null A?

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T/F: if subspace V is defined V=span{v1, …, vn} then {v1, …, vn} is a basis for V?

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T/F: any basis for R^n must have exactly n vectors.

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T/F: if Ax = b and b is not the zero vector then x is certainly not in the null space of A.

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T/F: if A is mxn and m > n then A^T may not have independent columns

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T/F: BAx must be in col(A)? Must be in col(B)?

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T/F: any matrix whose columns form a basis for R^n has pivots every row and every columns.

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T/F: Let A be mxn. Then Ax is in Col(A) for every x in R^n.

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If A is size mxn, and b is in Col(A) for every b in R^m, the the transformation T: R^n |-> R^m defined by T(x)=Ax is onto (surjective).

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T/F: let A be a 3x4 matrix. Then the transformation whose standard matrix is A cannot be onto.

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Counterexample: [[1, 0], [0, 1]] can��t wait until we get to eigenvectors unit this semester

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T/F: if Ax=x for some then A must not be invertible.

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This is a tough one for my students but if w were in fact a LC of u and v then c1u+c2v-w=0 for some weights c1 and c2 and thus the set {u, v, w} must be dependent. Are my followers familiar with contrapositive arguments?

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T/F: if {u, v, w} is a linearly independent set of vectors then w is certainly not a linear combination of u and v

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counterexample: A=[[1 0], [0, 0]] and b = [0, 1]

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T/F: suppose Ax=b is inconsistent for some b where A is size mxn. We must therefore have m>n.

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This is true because if the cols are LI (call them a1, …, an) then the only solution to
c1a1+…+cnan=0
Is the trivial one (this is the textbook definition of LI)
And so Ax=0 has only the trivial solution x=0.

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T/F: If the columns of A are linearly independent then the null space of A contains only the zero vector?

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This is true because the range of the transformation is nothing but all of the linear combinations of the columns of A, all of which are vectors in R^m.

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If A is size mxn then the range of the transformation T(x)=Ax is a subset of R^n?

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Gonna go on a solutions release spam on some recent tweets