lin ex2

Let H be a subspace of a vector space V. An indexed set of vectors B={b1,...,bp} in V is a basis for H if

(i) B is a linearly independent set, and (ii) the subspace spanned by B coincides with H; that is, H=Span{b1,...,bp}

A vector space is a nonempty set V of objects, called vectors, on which are defined two operations, called addition and multiplication by scalars (real numbers), subject to the ten axioms (or rules) listed below. 1 The axioms must hold for all vectors u, v, and w in V and for all scalars c and d.

1. The sum of u and v, denoted by u+v is in V. 2. u+v = v+u 3. (u+v)+w = u+(v+w) 4. There is a zero vector 0 in V such that u + 0= u. 5. For each u in V, there is a vector −u in V such that u + (−u) = 0. 6. The scalar multiple of u by c, denoted by cu, is in V. 7. c (u+v) = cu + cv 8. (c+d) u = cu + du 9. c (du) = (cd) u. 10. 1u=u.

An eigenvector of an n×n matrix A is a nonzero vector x such that Ax=λx for some scalar λ.

A scalar λ lamda is called an eigenvalue of A if there is a nontrivial solution x of Ax=λx; such an x is called an eigenvector corresponding to λ. lamda

Is Col A= R5​? Explain your answer. Choose the correct answer and reasoning below.

A. ​Yes, because the column space of a 5×8 matrix is a subspace of set of real numbers R5. There is a pivot in each​ row, so the column space is 5-dimensional. Since any 5​-dimensional subspace of set of real numbers R 5 is set of real numbers R5​, Col A = R5.

An Inverse Formula Let A be an invertible n×nn×n n times n matrix. Then

A^−1=(1/det A)*adj A

Theorem 5 The Diagonalization Theorem

An n×n matrix A is diagonalizable if and only if A has n linearly independent eigenvectors. In fact, A=PDP−1, with D a diagonal matrix, if and only if the columns of P are n linearly independent eigenvectors of A. In this case, the diagonal entries of D are eigenvalues of A that correspond, respectively, to the eigenvectors in P.

Theorem 6

An n×n matrix with n distinct eigenvalues is diagonalizable.

Is Nul A=ℝ3​? Explain your answer. Choose the correct answer and reasoning below.

B. ​No, because the null space of a 5 ×8 matrix is a subspace of set of real numbers R8. Although dim Nul A=3​, it is not strictly equal to set of real numbers R3 because each vector in Nul A has eight components. Each vector in set of real numbers R3 has three components.​ Therefore, Nul A is isomorphic to set of real numbers R3​, but not equal.

The column space of an m×n matrix A, written as Col A, is the set of all linear combinations of the columns of A. If A=[a1⋯an]

Col A=Span{a1,...,an}

t/f : The column space of​ A, Col(A), is the set of all solutions of Ax=b.

False because ​Col(A)equals=​{b : b=Ax for some x in ℝn​}

If A is a 2×2 matrix, the area of the parallelogram determined by the columns of A is |det A|

If A is a 3×3 matrix, the volume of the parallelepiped determined by the columns of A is |det A|.

rank theorem

If a matrix A has n columns, then rank A + dim Nul A = n

thoerem

If a vector space V has a basis of n vectors, then every basis of V must consist of exactly n vectors.

Theorem The Invertible Matrix Theorem (continued)

Let A be an n×n matrix. Then A is invertible if and only if: s. The number 0 is not an eigenvalue of A. t. The determinant of A is not zero.

basis theorem

Let H be a p-dimensional subspace of Rn. Any linearly independent set of exactly p elements in H is automatically a basis for H. Also, any set of p elements of H that spans H is automatically a basis for H.

therom

The eigenvalues of a triangular matrix are the entries on its main diagonal.

theorem

The null space of an m×n matrix A is a subspace of Rn. Equivalently, the set of all solutions to a system Ax=0 of m homogeneous linear equations in n unknowns is a subspace of Rn.

theorem

The null space of an m×n matrix A is a subspace of Rn. Equivalently, the set of all solutions to a system Ax=0Ax=0 eh x equals 0 of m homogeneous linear equations in n unknowns is a subspace of Rn.

theorem

The pivot columns of a matrix A form a basis for Col A.

rank

The rank of A is the dimension of the column space of A.

The set M2×2 of all 2 × 2 matrices is a vector​ space, under the usual operations of addition of matrices and multiplication by real scalars. Determine if the set H of all matrices of the form [a b] is a subspace of M2×2 [0 d]

The set H is a subspace of M2×2 because H contains the zero vector of M2×2​, H is closed under vector​ addition, and H is closed under multiplication by scalars.

Determine if the given set is a subspace of set of prime numbers P8. Justify your answer. The set of all polynomials of the form p​(t)=at^8​, where a is in set of real numbers R. Choose the correct answer below.

The set is a subspace of set of prime numbers P8. The set contains the zero vector of set of prime numbers P8, the set is closed under vector​ addition, and the set is closed under multiplication by scalars.

Determine if the given set is a subspace of set of prime numbers P^n. Justify your answer. The set of all polynomials in set of prime numbers P^n such that p​(0) = 0

The set is a subspace of set of prime numbers P^n because the set contains the zero vector of set of prime numbers P^n, the set is closed under vector​ addition, and the set is closed under multiplication by scalars.

row space of A and is denoted by Row A

The set of all linear combinations of the row vectors

t/f: If B is an echelon form of a matrix​ A, then the pivot columns of B form a basis for Colnbsp A.

The statement is false because the columns of an echelon form B of A are not necessarily in the column space of A.

t/f: The standard method for producing a spanning set for Nul A sometimes fails to produce a basis for Nul A.

The statement is false because the method always produces an independent set.

t/f: A linearly independent set in a subspace H is a basis for H.

The statement is false because the subspace spanned by the set must also coincide with H.

Row operations preserve the linear dependence relations among the rows of A.

The statement is false. Row operations may change the linear dependence relations among the rows of A.

t/f: If B is any echelon form of​ A, then the pivot columns of B form a basis for the column space of A.

The statement is false. The columns of an echelon form B of A are often not in the column space of A.

The row space of AT is the same as the column space of A.

The statement is true because the rows of AT are the columns of (A^T)^T = A.

t/f: If a finite set S of nonzero vectors spans a vector space​ V, then some subset of S is a basis for V.

The statement is true by the Spanning Set Theorem.

t/f: A basis is a linearly independent set that is as large as possible.

The statement is true by the definition of a basis.

If A and B are row​ equivalent, then their row spaces are the same.

The statement is true. If B is obtained from A by row​ operations, the rows of B are linear combinations of the rows of A and​ vice-versa.

If B is a basis for a subspace​ H, then each vector in H can be written in only one way as a linear combination of the vectors in B. Choose the correct answer below.

The statement is true. Suppose B={v 1 ...,vp} and x is a vector in H that can be generated two ways.​ Say, x=c1v1+...+cpvp and x= d1v1+...+dpvp​, then 0= x−x = (c1 - d1) + (cp -dp). ​Therefore, cp=dp and x can only be generated in one way.

The dimension of the null space of A is the number of columns of A that are not pivot columns.

The statement is true. The dimension of Nul A equals the number of free variables in the equation Ax=0.

t/f : The column space of an m × n matrix is in set of real numbers Rm.

True because the column space of an m ×n matrix A is a subspace of set of real numbers Rm

t/f : A null space is a vector space.

True because the null space of an m x n matrix A is a subspace of set of real numbers Rn

t/f : The null space of​ A, Nul(A), is the kernel of the mapping x↦Ax.

True, the kernel of a linear transformation​ T, from a vector space V to a vector space​ W, is the set of all u in V such that T(u​)=0. ​Thus, the kernel of a matrix transformation T(x​)=Ax is the null space of A.

t/f: The set of all solutions of a homogeneous linear differential equation is the kernel of a linear transformation.

True, the linear transformation maps each function f to a linear combination of f and at least one of its​ derivatives, exactly as these appear in the homogeneous linear differential equation.

Theorem 3 Properties of Determinants

a. A is invertible if and only if det A≠0. b. det AB=(det A)(det B). c. det AT=det A.. d. If A is triangular, then det A is the product of the entries on the main diagonal of A. e. A row replacement operation on A does not change the determinant. A row interchange changes the sign of the determinant. A row scaling also scales the determinant by the same scalar factor.

A subspace of a vector space V is a subset H of V that has three properties

a. The zero vector of V is in H. 2 b. H is closed under vector addition. That is, for each u and v in H, the sum u + v is in H. c. H is closed under multiplication by scalars. That is, for each u in H and each scalar c, the vector c u is in H.

A square matrix A is invertible

if and only if det A≠0.

The column space of an m×n matrix A

is a subspace of Rm.

Theorem The Invertible Matrix Theorem (continued) -Let A be an n×n matrix. Then the following statements are each equivalent to the statement that A is an invertible matrix.

m. The columns of A form a basis of Rn. n. Col A=Rn o. dim Col A=n p. rank A=n q. Nul A={0} r. dim Nul A=0

rank

of a matrix A, denoted by rank A, is the dimension of the column space of A. rank # = # of pivot columns

dimension

of a nonzero subspace H, denoted by dim H, is the number of vectors in any basis for H. The dimension of the zero subspace {0}{0} the set 0 end set is defined to be zero

null space

the set of all solutions of the homogeneous equation Ax=0

If v1,...,vp are in a vector space V,

then Span {v1,...,vp} is a subspace of V.

If A and B are n×n matrices,

then det AB=(det A)(det B).

if two rows of A are interchanged to produce B,

then det B= −det A.

. If a multiple of one row of A is added to another row to produce a matrix B

then det B=det A.

If one row of A is multiplied by k to produce B,

then det B=k⋅det A.

If A is an n×n matrix,

then detA^T=det A.

rank A+dim Nul A=n

{number of pivot columns}+{number of non pivot columns}={number of columns}

t/f: The range of a linear transformation is a vector space.

​True, the range of a linear transformation​ T, from a vector space V to a vector space​ W, is a subspace of W.