Linear Algebra Quiz true/false

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A vector space must contain at least two vectors.

False

Every basis of P4 contains at least one polynomial of degree 3 or less.

False

Every linearly independent subset of a vector space V is a basis for V.

False

Every subset pf a vector space V that contains the zero vector in V is a subspace of V.

False

If A is a 3 X 3 matrix and B is obtained from A by adding 5 times the first row to each of the second and third rows, then det(B) = 25det(A).

False

If A is a n X n matrix and B is obtained from A by multiplying each row of A by its row number, then det(B0 = ((n(n+1)/2)det(A))

False

If A is a square matrix, then A = P(b1 ->b2) for some bases B1 and B2 for R^n.

False

If A is square and Ax = b is inconsistent for some vector b, then the nullity of A is zero.

False

If E is a an m x m elementary matrix and A is an m x n matrix, then the column space of EA is the same as the column space of A.

False

If R is the reduced row echelon form of A, thent hose column vectors of R that contain the leading 1's form a basis for the column space of A.

False

If V = span{v1,...,Vn}, then {v1,...,Vn} is a basis for V.

False

If V is a subspace of R^n and W is a subspace of V, then W(perpendicular) is a subspace of V(perpendicular).

False

If each vector in B2 is a scalar multiple of some vector in B1, then P(b1->b2) is a diagonal matrix

False

If rank(A^t) = rank(A), then A is square.

False

If u is a vector and k is a scalar such that ku = 0, then it must be trust that k = 0.

False

In the vector space F(−∞,∞) any function whose graph passes through the origin is a zero vector.

False

The column space of a matrix A is the set of solutions of Ax = b.

False

The functions f1, and f2 are linearly dependent if there is a real number x such that k1f1(x) + k2f2(x) = 0 for some scalars k1 and k2.

False

The kernel of a matrix transformation Ta: R^n --> R^m is a subspace of R^m.

False

The polynomials x-1, (x-1)^2, and (x-1)^3 span P3

False

The set of 2 x 2 matrices that contain exactly two 1's and two 0's is a linearly independent set in M22.

False

The set of positive real numbers is a vector space if vector addition and scalar multiplication are the usual operations of addition and multiplication of real numbers.

False

The span of a single vector in R^2 is a line.

False

The span of two vectors in R^3 is a plane

False

The union of any two subspaces of a vector space V is a subspace of V.

False

There are at least two distinct three-dimensional subspaces of P2.

False

There are only three distinct two-demnsional subspaces of P2.

False

There is a set of 11 vectors that span R^17.

False

There is an invertible matrix A and a singular matrix B such that the row spaces of A and B are the same.

False

the solution set of a consistent linear system Aa = b of m equations in n unknowns is a subspace of R^n.

False

For every square matrix A and every scalar c, it is true that det(CA = cdet(A).

False.

For all square matrices A and B, it is true that det(A + B) = det(A) + det(B)

False. Only works for multiples not sums.

The determinant of a lower triangular matrix is the sum of the entries along the main diagonal.

False. Its the multiple of all the entries along the main diagonal.

A set containing a single vector is linearly independent.

False. This is true only if the vector is a nonzero vector.

How do you know if it forms a basis?

If it spans and is linearly independent.

You put a matrix through RREF [1001][010(-1)][0011][0000]. Is this linearly dependent or Linearly independent.

Linearly Dependent.

You put a matrix through RREF [1000][0100][0010][0001]. Is this linearly dependent or Linearly independent.

Linearly independent.

A matrix with linearly independent row vectors and linearly independent column vectors is square.

True

Every linearly independent set of five vectors in R^5 is a basis for R^5.

True

Every linearly independent set of vectors in R^n is a contained in some basis for R^n.

True

Every set of fiver vectors that spans R^5 is a basis for R^5.

True

Every set of vectors that spans R^n contains a basis for R^n.

True

If A is a 3 X 3 matrix and B is obtained from A by multiplying the first column by 4 and multiplying the third column 3/4, then det(B) = 3det(A).

True

If E is a an m x m elementary matrix and A is an m x n matrix, then the null space of EA is the same as the null space of A.

True

If P(b1->b2) is a diagonal matrix, then each vector is a scalar multiple of some vector in B1.

True

If V1,...,Vn are linearly deendent nonzero vectors, then at least on vector Vl is a unique linear combination of v1,....,vk-1.

True

If the sum of the second and fourth row vectors of a 6 X 6 matrix A is equal to the last row vector, then det(A) = 0.

True

No Linearly independent set contains the zero vector.

True

The coordinate vector of a vector x in R^n relative to the standard basis for R^n is x.

True

The intersection of any two subspaces of a vector space V is a subspace of V.

True

The nullity of a square matrix with linearly dependent rows is at least one.

True

The span of a nonempty set S of vectors in V is the smallest subspace of V that contains S.

True

The three polynomials (x-1)(x+2), x(x+2), and x(x-1) are linearly independent.

True

There is a basis for M22 consisting of invertible matrices.

True

There is a set of 17 linearly independent vectors in R^17.

True

There is no 3 x 3 matrix whose row space and null space are both lines in 3-space

True

if A has a size n x n and In, A, A3,...A^n^2 are distinct matrices, then {In, A,A^2,....An^2} is a linearly dependent set.

True

if A is a 4 X 4 matrix and B is obtained from A by interchanging the first two rows and then interchanging the last two rows, then det(B) = det(A).

True

If E is a an m x m elementary matrix and A is an m x n matrix, then the row space of EA is the same as the row space of A.

True.

What is the dimension of the vector space of all diagonal n x n matrices

n

What is the dimension of the vector space of all symmetric n x n matrices

n(n+1)/2

What is the dimension of the vector space of all upper triangle n x n matrices,

n(n+1)/2


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