Vector Spaces III Lowersixth Science Further Mathematics

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Core concepts for vectors, spaces, and transformations.

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YAKILI LMS

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Vector Spaces III Lowersixth Science Further Mathematics


Froggy Jumps Vector Spaces III Lowersixth Science Further MathematicsOnline version Core concepts for vectors, spaces, and transformations. by YAKILI LMS 1 What defines linear dependence among vectors? a All vectors are zero b All vectors are equal c A nontrivial combination equals zero 2 What is a basis of a vector space? a A set of independent vectors spanning the space b A set of vectors with a common direction c A single vector that repeats 3 What is the dimension of a vector space? a Number of vectors in a basis b Largest eigenvalue c Number of vectors in the space 4 If v1, v2 are independent, what is their span? a All linear combinations of v1 and v2 b Only v1 and v2 themselves c Only zero vector 5 What is a linear transformation? a A map preserving addition and scalar multiplication b A map that inverts vectors c A map preserving only addition 6 Matrix of a linear transformation depends on what? a The determinant only b The dimension only c Choice of basis for domain and codomain 7 What does the matrix A represent in T(x) = Ax? a The inverse transformation b The eigenvector c The linear transformation relative to bases 8 Eigenvalues satisfy what equation? a Det(A) = 0 b Aλ = I c Det(A - λI) = 0 9 Eigenvectors correspond to which relation? a Av = λv for some nonzero v b Av = v c Av = 0 for all v 10 What is an eigenvector of a matrix A? a A nonzero vector v with Av = λv b A vector with determinant 1 c Any vector mapped to zero 11 How many eigenvalues can a 2x2 matrix have (counting multiplicity)? a Infinite b Two c One 12 If A is diagonalizable, what can be said about A? a A has zero determinant b A is always symmetric c There exists a P such that P^{-1}AP is diagonal 13 What is the eigenvector for a multiple of the identity matrix I? a Only the zero vector b Only vectors perpendicular to I c Any nonzero vector 14 What does the trace of A equal when A has eigenvalues λ1, λ2? a Sum of eigenvalues b Determinant of A c Product of eigenvalues 15 If v is an eigenvector for eigenvalue λ, what is Av? a v/λ b Av = v c λv 16 What is the null space of a matrix A? a Set of eigenvectors only b Set of x with Ax = 0 c Set of x with Ax = x 17 Rank-Nullity Theorem relates to what? a Trace equals nullity b Dimension of domain = rank(A) + nullity(A) c Determinant equals rank 18 What is a linear transformation’s kernel? a Set of vectors mapped to zero b Set of vectors mapped to identity c Set of eigenvectors 19 What is a linear transformation’s image? a Determinant of T b Set of all outputs T(v) c Set of all inputs v 20 What does a 3x3 matrix represent in R^3? a A scalar function b A vector in R^3 c A linear map from R^3 to R^3 21 Which operation preserves linear independence? a Multiplying by zero b Adding dependent vectors c Applying an invertible matrix 22 What is a basis for R^2? a One vector b Two independent vectors in R^2 c Three dependent vectors 23 What is the dimension of span{e1, e2} in R^2? a 2 b 1 c 3 24 What is a column space? a Span of the matrix columns b Span of the rows c The null space itself 25 If columns of A are dependent, what is rank(A)? a Equal to number of columns b Zero c Less than number of columns 26 Eigenvectors corresponding to distinct eigenvalues are a Zero b Linearly independent c Identical 27 What is a diagonal matrix D? a Only diagonal entries nonzero b Nonzero off-diagonal entries c All zeros only 28 What does eigenvalue decomposition require? a A is symmetric only b A is singular c A is diagonalizable 29 What is a nullity of A? a Sum of eigenvalues b Dimension of row space c Dimension of the null space 30 A linear transformation preserves a Distance only b Angle only c Vector addition and scalar multiplication 31 Which set is a subspace? a Any subset of vectors b Nonempty, closed under addition and scalar mult c Only the zero vector 32 What is the image of a basis under a linear map? a Always the zero vector b Always equal to input c The set of their images spanning the target 33 What is the determinant zero indicating? a Matrix is singular b Matrix is diagonal c Matrix invertible 34 What is a similarity transformation? a Transposing A only b P^{-1}AP = B for some A,B c A new matrix equal to A 35 What is the characteristic polynomial? a Trace(A) - λ b Det(A + λI) c Det(A - λI) 36 What can you say about eigenvectors of a triangular matrix? a Eigenvectors are on diagonal b Eigenvalues are on the diagonal c No eigenvalues exist 37 How do you test independence of a set? a See if they are all equal b Check length only c Solve c1v1 + c2v2 + ... = 0 38 What is a homogeneous system? a Right-hand side equals identity b All constant terms are zero c All variables are zero 39 What does a basis enable you to do? a Represent any vector uniquely b Create zero vectors only c Change dimension arbitrarily 40 What is the rank of a zero matrix? a n b 1 c 0 41 What is a linear transformation’s matrix relative to bases? a Determinant only b Matrix representation mapping coordinates c Scalar value 42 What does a full rank imply? a Independent columns equal dimension b Dependent columns c Zero determinant 43 What is the eigenvector condition for A in R^n? a Av = 0 for all v b Av = λv for some λ c Av = v 44 What is a linear combination? a Product of vectors b Sum of scalars times vectors c Difference of vectors 45 What is the image of the standard basis under a map? a The rows of the map b The column vectors of the map c Always zero 46 What is an invariant subspace? a Only zero vector b Mapped to itself under T c Mapped outside itself 47 What is a linear independence test? a Nontrivial combination equals zero b All coefficients equal one c Only trivial combination equals zero 48 What is a 2x2 eigenvalue problem example? a Solve Av = I b Compute A^2 = I c Solve det(A - λI) = 0 49 What is a basic property of eigenvectors for distinct λ? a They must be orthogonal b They are identical c They are linearly independent 50 What does diagonalization require? a A has enough eigenvectors to form a basis b Matrix must be symmetric only c Determinant must be zero

Vector Spaces III Lowersixth Science Further Mathematics

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Vector Spaces III Lowersixth Science Further MathematicsOnline version

by YAKILI LMS

What defines linear dependence among vectors?

What is a basis of a vector space?

What is the dimension of a vector space?

If v1, v2 are independent, what is their span?

What is a linear transformation?

Matrix of a linear transformation depends on what?

What does the matrix A represent in T(x) = Ax?

Eigenvalues satisfy what equation?

Eigenvectors correspond to which relation?

What is an eigenvector of a matrix A?

How many eigenvalues can a 2x2 matrix have (counting multiplicity)?

If A is diagonalizable, what can be said about A?

What is the eigenvector for a multiple of the identity matrix I?

What does the trace of A equal when A has eigenvalues λ1, λ2?

If v is an eigenvector for eigenvalue λ, what is Av?

What is the null space of a matrix A?

Rank-Nullity Theorem relates to what?

What is a linear transformation’s kernel?

What is a linear transformation’s image?

What does a 3x3 matrix represent in R^3?

Which operation preserves linear independence?

What is a basis for R^2?

What is the dimension of span{e1, e2} in R^2?

What is a column space?

If columns of A are dependent, what is rank(A)?

Eigenvectors corresponding to distinct eigenvalues are

What is a diagonal matrix D?

What does eigenvalue decomposition require?

What is a nullity of A?

A linear transformation preserves

Which set is a subspace?

What is the image of a basis under a linear map?

What is the determinant zero indicating?

What is a similarity transformation?

What is the characteristic polynomial?

What can you say about eigenvectors of a triangular matrix?

How do you test independence of a set?

What is a homogeneous system?

What does a basis enable you to do?

What is the rank of a zero matrix?

What is a linear transformation’s matrix relative to bases?

What does a full rank imply?

What is the eigenvector condition for A in R^n?

What is a linear combination?

What is the image of the standard basis under a map?

What is an invariant subspace?

What is a linear independence test?

What is a 2x2 eigenvalue problem example?

What is a basic property of eigenvectors for distinct λ?

What does diagonalization require?