True/False Math 33A Midterm 1

A system of 4 linear equations in 3 unknowns is always inconsistent

FALSE; a fourth equation can be consistent with the first three equations and give the system just one answer

If A is a 3x4 matrix and vector v is in R4 then vector Av is in R3

TRUE; the product of a 3x4 matrix and 4x1 matrix is 3x1

There exists a system of 3 linear equations with 3 unknowns that has 3 solutions

FALSE; a system of linear equations can only have 1 solution, infinite solutions, or no solutions

Vector [123] is a linear combination of vectors [456] and [789]

TRUE; the system is consistent with x=2 and y=-1

If A and B are any two 3x3 matrices of rank 2, then A can be transformed into B by means of elementary row operations

FALSE; A can only be transformed to B if ref(A)=rref(B)

There exists a 3x4 matrix with rank 4

FALSE; there are only 3 rows so there can only be rank 3

If matrix A is in reduced row echelon form, then at least one of the entries in each column must be 1

FALSE; a matrix can have a column with no leading ones

If v and w are vectors in R4, then v must be a linear combination of v and w

TRUE; a linear combination of v and w would produce v if v=1v+0w

A linear equation with fewer unknowns than equations must have infinitely many solutions or none

FALSE; it is possible for every equation to correspond to one answer

If vector u is a linear combination of vectors v and w then w must be a linear combination of u and v

FALSE; a linear combination is u=av+bw and if b=0 then bw =\ u-av

The matrix [5 6]

[-6 5]

represents a rotation combined with a scaling

TRUE; the matrix is in the form for rotations, which is [a -b]

[b a] where a=5 and b=-6

If AB=In for two non matrices A and B then A must be the inverse of B

TRUE; the definition of an invertible matrix is that AB=In

Matrix [k -2]

[5 k-6]

is invertible for all real numbers K

True because the det(A) will never equal zero since 0=k^2-6k+10

There exists an invertible nxn matrix with two identical rows

FALSE; if a matrix had two identical rows the det(A)=ab-ab=0

If A^2=In then matrix A must be invertible

TRUE; a condition for an invertible matrix is that AA-1=In and A-1A=In so if A^2=In this must mean that A=A-1

The formula (A^2)-1 = (A-1)^2 holds for all invertible matrices A

TRUE; AA is invertible therefore (AA)-1=A-1A-1=(A-1)^2

If A is a 3x4 matrix and B is a 4x5 matrix then AB will be a 5x3 matrix

FALSE; AB will be a 3x5 matrix

If A and B are two 4x3 matrices such that Av=Bv for all vectors v in R3, then matrices A and B must be equal

TRUE; only one unique matrix can satisfy this condition

If v1, v2, ..., vm and w1, w2, ..., wm are any two bases of a subspace V of R10 then n must equal m

TRUE; bases of a given subspace have the same number of elements

The span of vectors v1, v2,..,vm consists of all linear combinations of vectors v1, v2, vn

TRUE; the span contains all linearly independent vectors, which means all other vectors can be formed by a linear combination of the span

If matrix A is similar to matrix B, and B is similar to C, then C must be similar to A

TRUE; similar matrices have the transitive property

If vectors v1, v2, v3, and v4 are linearly independent, then vectors v1, v2, and v3 must be linearly independent as well

TRUE; taking away one vector doesn't change that the rest of the vectors are linearly independent

If the kernel of matrix A consists of the zero vector only, then the column vectors of A must be linearly independent

TRUE; if Ker(A) = {0}, then the matrix is invertible and another condition for invertibility is linearly independent columns

If two nonzero vectors are linearly dependent, then each of them is a scalar multiple of the other

TRUE; if just 2 vectors are linearly dependent, this means you have to be able to multiply one by a constant to get the other

If v1, v2, vn are linearly independent vectors in Rn then they must form a basis of Rn

TRUE; this is the definition of a basis

If a subspace V of Rn contains none of the standard vectors e1, e2, en then V consists of the zero vector only

FALSE; a subspace doesn't have to contain standard basis vectors

If A and B are invertible nxn matrices then AB must be similar to BA

TRUE; similar matrices satisfy AS=SB and ABS=SBA. If S=B-1 then AB(B-1)=AI and (B-1)(B)A=IA which both equal A

Matrix In is similar to 2In

FALSE; this doesn't satisfy the equation AS=SB

If AB=0 for two 2x2 matrices A and B then BA must be the zero matrix as well

FALSE; matrix multiplication is not commutative

There exists a 2x2 matrix A such that Im(A)=Ker(A)

TRUE; for the zero matrix Im(A)=R2 and Ker(A)=R2

Let A be a n x m matrix and let b exist in Rn. If Ax=b has a unique solution then A must be a square matrix

FALSE

Let A be an nxn matrix and let b exist in Rn. If Ax=b has a unique solution, then Ax=0 also has a unique solution

TRUE

If A and B are both nxn matrices then AB=BA

FALSE; matrices don't commute

If A and B are both nxn matrices and AB=In then B is invertible with B-1=A

TRUE

The function T: R^2 to R^2 T([x1 x2]) = [x2 1] is linear

FALSE

The system Ax=b is inconsistent iff rref(A) contains a row of zeros

FALSE

If A^2 = A for an invertible nxn matrix A, then A must be In

TRUE

If matrix A commutes with matrix B, and B commutes with matrix C, then A must commute with C

FALSE

If A and B are invertible nxn matrices then ABA-1B-1 = In

FALSE

Every vector in R4 is a linear combination of e1, e2, e3, and e4

TRUE

If A is a 3x4 matrix of rank 3, then the system Ax = [1 2 3] has infinitely many solutions

TRUE