Let A be an m × n matrix. Prove that every vector x in Rn can be written in the form x = p+ u, where p is in Row A and u is in Nul A. Also, show that if the equation Ax = b is consistent, then there is a unique p in Row A such that Ap =b.

Q: If C is a real square matrix, then C² + I (I is the identity matrix ) is non-singular. - The vector…

A: Find the dimensions of U and V.  Then use the result if two vector spaces have same dimension then…

Q: Show that if the entries in each row of an n × n matrix A add up to zero, then the column vectors of…

A: linearly dependent means det of A is zero

Q: 27. Consider an n × m matrix A, a vector v in R", and a vector w in R". Show that (Av) · w = i · (A™…

A:

Q: Let x1, x2, and x3 be linearly independent vectors in R4 and let A be a nonsingular 4 × 4 matrix.…

A: Given that:

Q: For the matrix A in , use the Cayley ­Hamilton Theorem to compute A2, A3, and A4 by expressing each…

A: Given matrix is

Q: Let A be an m×n matrix. Explain why the followingare true. Any vector b ∈ Rm can be uniquely written…

A: For  matrix A, N(A) is the null space of A. R(A) is the column space of the matrix A and R(AT) is…

Q: Show that A + AT is symmetric for any square matrix A.

A:

Q: Let A be an (n x n) matrix and za column vector. If AA = I then Ax|| = ||r||. %3D %3D O True O False

A:

Q: Let A be a 2 × 2 matrix with linearly independentcolumn vectors a1 and a2. If a1 and a2 are usedto…

A:

Q: Show that the column vectors of following matrix A are linearly independent:

A: the given question is taken from the matrix in which we have to find show that the given matrix A is…

Q: Show that if A is a mxn matrix and rank(A)=m then Ax=b has a solution for any vector b in Fm (Hint:…

A:

Q: Let A be a nonsingular 4 x 4 matrix such that A = -2, then AT adj(A ) Select one: O True O False 17

A: properties of the determinant

Q: Suppose that A and B are matrices having n rows. Prove that M(A|B) = (MA|MB) for any m ×n matrix M.

A:

Q: Let ~v and ~w be column vectors in R^n, and let A be an n×n matrix. Prove that, if {A (~v), A (~w)}…

A:

Q: Construct a 3x3 matrix A, with nonzero entries, and a vector b in R such that b is not in the set…

A: using the concept of vector space , approches to check one by one option.

Q: Let A be a square matrix. Show that (a) A + A# is Hermitian, (b) A – A# is skew-Hermitian, (c) A =…

A: Explanation of the answer is as follows

Q: Prove that, for m X n matrices A and B, rank (A + B) <=rank(A) + rank(B

A:

Q: Prove that B is row equivalent to A if and only if there exists a nonsingular matrix M such that B =…

A:

Q: Let D = P'AP be a diagonal matrix, wher %3D Pesell thet if D:

A: Given, D=P-1AP be a diagonal matrix, where P is an orthogonal matrix. To prove that A is symmetric.

Q: Show that if A is an orthogonal matrix, then det(A) = ±1.

A: Given If A is an orthogonal matrix. To show det(A) = ±1

Q: Let A be an m × n matrix and u a vector in Rn. Suppose that Au = 0 (where 0 is the zero vector in…

A:

Q: If A is an n × n matrix, show that (Ak )T = (AT )k for any positive integer k.

A: Matrix: We will solve it by the method of induction:(Ak)T=(AT)k case 1: k=1case 2: k=ncase 3: k=n+1

Q: Let A be an n × n matrix in which the entries of each row sum to zero. Find det(A).

A:

Q: Let Bj denote the matrix obtained by replacing thejth column of the identity matrix with a vectorb =…

A: COnsider the system of equations:IX =…

Q: Let A be a m xn matrix with columns c1, ... , en. Prove that {c1, ..., cn} is independent if and…

A: To prove  c1,c2,...cn is independent if and only if null A=0

Q: Show that if the non-zero vectors s(1), s(2),..., s(m) are conjugate to a positive definite matrix…

A: We are given that the non-zero vectors s(1), s(2), ..., s(n) are conjugate to a positive definite…

Q: Let A be an m×n matrix. Explain why the following are true. Any vector x in Rn can be uniquely…

A: Given : A is an m×n matrix. To show : a) Any vector x in ℝn can be uniquely written as a sum y+z,…

Q: Let A be a square matrix such that A2 = 0. Prove that (Af)2 = 0

A: Given that A is a square matrix and A2=0. In order to prove (At)2=0 following theorem are used.…

Q: Let A and B be matrices of size p × m and n × q , respectively. Assume that mn = pq . Define R: Mmn…

A: Matrix A is p×m and B is n×q In a m×n matrix number of element will be m.n R is defined as,…

Q: () (:) Find the matrix A satisfying A and A and write the SVD of A in 2. vector form. [Hint: Start…

A: NOTE: According to guideline answer of first three subpart can be given, for other please ask in a…

Q: Let U be the row-echelon form of square matrix A. prove that if the first r columns of U are…

A:

Q: Show that if A is an orthogonal matrix, then det(A) = +1.

A: To prove, detA=±1

Q: Let A be an m x n matrix, and let b1,..., bk be vectors in R" such that the system b; has a solution…

A: This question related to linear algebra's topic linearly independent set.

Q: For a nonsingular matrix A and nonzero scalar k, show that (k A)−1 = 1 k A−1.

A: Given that A is nonsingular matrix. That is, A≠0. Since k is a non zero scalar, kA≠0. Therefore, kA…

Q: Let A = [aij] be a square matrix If B is obtained by adding a multiple of one row (column) of A to…

A:

Q: For a matrix A, define Col(A) to be the set consisting of all linear combinations of column vectors…

A:

Q: Give a formula for (ABx)T, where x is a vector and A and B are matrices of appropriate size. Choose…

A: To find (ABx)T.

Q: Show that if A is an n × n matrix, then det(AAT ) ≥ 0.

A: Given, A is a n x n matrix. Let AT be the transpose of A. To prove det (AAT) is non-negative.

Q: Show that is A is a square matrix, then A + AT is a symmetric matrix. You can use any matrix for A…

A:

Q: Let the entries of a matrix A = \aij be defined as aij 2i2 – i+j+ g(j), where g is a %3D function of…

A: Given Ais a symmetric matrix therefore aij=aji…

Q: let  A and  B be  square  matrices of order n  such that  Ax=Bx  and x belongs to Rn  then   find…

A: Ax = Bx implies  Ax - Bx = O So, (A - B)x = O where x belongs to Rn  So, x belongs to the null space…

Q: Prove the following statement: If the columns of A are linearly independent, then AT A is…

A: A is invertible if there exists a matrix A-1 such that AA -1=A -1A=I   The vectors v1,.......,vn are…

Q: Let A be an m × n matrix.(a) Describe the diagonal entries of AT A in terms of the columns of A.(b)…

A: (a) Describe the diagonal entries of AT A in terms of the columns of A.

Q: Let A = [a;;] be the n x n matrix defined by aji = k and a;; = 0 if i + j. Show that if B is any n x…

A: Given: A=aijn×n,     aii=k and aij=0 if i≠k. Let B be a square matrix of order 3 which can be…

Q: Let A = [aij] be a square matrix. a. If A has a zero row (column), then prove that det A = 0.

A: Let A = [aij] be a square matrix and If A has a zero row (column). We need to prove that det A = 0.…

Q: 20. Can every vector in R* be written as a linear combination of the columns of the matrix B above?…

A:

Q: Construct a nonzero 3 × 3 matrix A and a nonzero vector b such that b is in Col A, but b is not the…

A: We will construct A and put b as a linear combination of column vectors in A.

Q: let A be an mxn matrix and v and w vectors in Rn. If v and w were both solutions to Ax=0, show that…

A:

Q: Show that for matrices A, B E RXn, a vector w = ABx can be written as w = E,(b{ x)a;, %3D :D1 where…

A: Since, A,B∈ℝn×n. ∴ A, B have 'n' rows and 'n' columns with entries as real numbers. The given vector…

Q: Prove that if A is an orthogonal matrix, then so are AT and A−1.

A: We have to prove that if A is an orthogonal matrix, then so are AT and A−1. If A is an orthogonal…

Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.