Linear Algebra Quiz true/false
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
