LET Linear Algebra Review

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Essential concepts for the LET exam in linear algebra.

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Rosemarie Sullera

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LET Linear Algebra Review


LET Linear Algebra ReviewVersión en línea Essential concepts for the LET exam in linear algebra. por Rosemarie Sullera 1 What is the dimension of R^n? a 2n b n-1 c 0 d n 2 Which condition defines linear independence? a Any combination yields zero b Vectors must be orthogonal c Only the trivial combination yields the zero vector d All coefficients must be 1 3 If det(A) = 0, what is A? a Diagonal b Singular (not invertible) c Identity d Invertible 4 What is an eigenvalue λ of A? a λ = 0 only b Av = v c A is zero matrix d There exists v ≠ 0 with Av = λv 5 Rank of a matrix equals... a Dimension of its row space b Determinant value c Trace of A d Number of columns 6 What is the minimum number of vectors to span R^3? a 1 b 3 c 4 d 2 7 What does orthogonality mean for vectors u, v? a They have equal length b Their cross product is zero c They are parallel d Their dot product is zero 8 A linear transformation T satisfies... a T(cx) = cT(x) for all c b T(ax+by) = aT(x) + bT(y) c T(x) is always invertible d T(x+y) = T(x) + T(y) 9 Matrix is diagonalizable if... a It has a zero determinant b It is strictly upper triangular c It has n linearly independent eigenvectors d All eigenvalues are zero 10 Null space dimension satisfies which relation? a dim(N(A)) = rank(A) b dim(N(A)) = det(A) c dim(N(A)) = n + rank(A) d dim(N(A)) = n - rank(A) 11 What is the dimension of the vector space spanned by the columns of a matrix? a Eigenvalue b Determinant of the matrix c Rank of the matrix d Trace of the matrix 12 Which property do linearly independent vectors satisfy? a All linear combinations equal zero b They must be orthogonal c They are all unit vectors d Only the trivial linear combination gives zero 13 What is the determinant of a product of two matrices A and B? a Det(A) + Det(B) b Det(A) * Det(B) c Det(A)^2 + Det(B)^2 d Det(A+B) 14 Which transformation preserves the origin and linearity? a Nonlinear mapping b Isometry c Affine transformation d Linear transformation 15 What is an eigenvalue of a square matrix A? a Scalar λ such that Av = λv for some nonzero v b The rank of A c A vector v with Av = v d The trace of A 16 Which of the following is a basis concept? a A set with maximum length b A set of orthogonal vectors only c A set that spans the space and is linearly independent d A set with equal vectors 17 What does the rank of a matrix indicate? a Determinant value b Number of zero rows c Sum of eigenvalues d Maximum number of linearly independent rows or columns 18 Which operation factorizes a matrix into U and L with no pivoting? a SVD b Eigen decomposition c QR decomposition d LU decomposition 19 What is the null space of a matrix A? a Set of eigenvectors of A b Set of all solutions to Ax = 0 c Set of all x with Ax = x d Column space of A 20 In inner product spaces, what is the norm derived from the inner product? a The length of a vector, sqrt(v·v) b The area of a parallelogram c The determinant of v d The angle between vectors

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LET Linear Algebra Review

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LET Linear Algebra ReviewVersión en línea

por Rosemarie Sullera

What is the dimension of R^n?

Which condition defines linear independence?

If det(A) = 0, what is A?

What is an eigenvalue λ of A?

Rank of a matrix equals...

What is the minimum number of vectors to span R^3?

What does orthogonality mean for vectors u, v?

A linear transformation T satisfies...

Matrix is diagonalizable if...

Null space dimension satisfies which relation?

What is the dimension of the vector space spanned by the columns of a matrix?

Which property do linearly independent vectors satisfy?

What is the determinant of a product of two matrices A and B?

Which transformation preserves the origin and linearity?

What is an eigenvalue of a square matrix A?

Which of the following is a basis concept?

What does the rank of a matrix indicate?

Which operation factorizes a matrix into U and L with no pivoting?

What is the null space of a matrix A?

In inner product spaces, what is the norm derived from the inner product?