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linear algebra
Recall the Rank Theorem, which states that if A is an mxn matrix, then...
Linear algebra: By the Rank theorem rank A+dimNul=n does rank=the dimension of the set? Please follow...
Linear algebra: By the Rank theorem rank A+dimNul=n does rank=the dimension of the set? Please follow the comment Example 1 0 0 0 0 1 1 0 0 0 0 1 The rank A=3 and dimension = 3????
Anton Chapter 4, Section 4.8, Supplementary Question 01 Find the rank and nullity of the matrix;...
Anton Chapter 4, Section 4.8, Supplementary Question 01 Find the rank and nullity of the matrix; then verify that the values obtained satisfy Formula (4) in the Dimension Theorem. [1 4 6 5 8] 3 -4 2 -1 -40 1-1 0 -2 -1 8 [ 4 7 15 11 -4 A = 1 Click here to enter or edit your answer rank(A) = Click here to enter or edit your answer nullity(A) = Click here to enter or edit your...
Linear Algebra 1. One of the most important applications of linear algebra to electronics is to analyze electronic circ...
Linear Algebra 1. One of the most important applications of linear algebra to electronics is to analyze electronic circuits that cannot be described using the rules for resistors in series or parallel such as the one shown figure in below. The goal is to calculate the current flowing in each branch of the circuit or to calculate the voltage at each node of the circuit. This circuit is called Loop Current Analysis of Electric Circuits. In this circuit and number...
About linear algebra,matrix; 2. (a) Use Octave as a Calculator to answer this question. Suppose that A and B are two 8 x 9 matrices. The (i.j)-entry of the matrix B is given by i *j -1. The (i. j)-en...
About linear algebra,matrix;
2. (a) Use Octave as a Calculator to answer this question. Suppose that A and B are two 8 x 9 matrices. The (i.j)-entry of the matrix B is given by i *j -1. The (i. j)-entry of the matrix A equals 0 if i + j is divisible by 5 and equals the (i,j)-entry of the matrix B otherwise. i. What are the rank and nullity of matrices A and B? ii. Is vector u 9,64,-71,...
CHALLENGE ACTIVITY 5.5.1: Rank and nullity of a linear transformation. Jump to level 1 1 2...
CHALLENGE ACTIVITY 5.5.1: Rank and nullity of a linear transformation. Jump to level 1 1 2 Let T:U + V be a linear transformation. Use the rank-nullity theorem to complete the information in the table below. > -D- D - D + 3 4 P5 6 Pg Ex: 5 Pn Ex: n+2 U dim(U) rank(T) nullity(T) 4 Ex: 5 6 Ex: n+2 7 Ex: 5 2. Check Next Feedback?
Advanced Linear Algebra (bonus problem) 1. (This question guides you through a different proof of part of the Decomposi...
Advanced Linear Algebra (bonus problem)
1. (This question guides you through a different proof of part of the Decomposition Theorem. So you are not allowed to use the Decomposition Theorem when answering this question.) Let F be a field and V an n-dimensional F-vector space for n > I. Let θ E End(V) be a linear transformation and α E F an eigenvalue of. Recall that the generalised α-eigenspace of θ is a) Suppose that 0 υ Ε να and...
The Cayley-Hamilton Theorem states that a matrix satisfies its characteristic equation. For example, the characteristic equation...
The Cayley-Hamilton Theorem states that a matrix satisfies its characteristic equation. For example, the characteristic equation of the matrix shown below is as follows. 1-3 A = 12 - 61 + 11 = 0 and by the theorem you have A2 - 64 + 1112 = 0 2 5 Demonstrate the Cayley-Hamilton Theorem for the matrix A given below. 0 5 -1 -1 3 1 0 0 1 STEP 1: Find and expand the characteristic equation. STEP 2: Compute the...
The Cayley-Hamilton Theorem states that a matrix satisfies its characteristic equation. For example, the characteristic equation...
The Cayley-Hamilton Theorem states that a matrix satisfies its characteristic equation. For example, the characteristic equation of the matrix shown below is as follows. --1:: 22 - 61 + 11 = 0 and by the theorem you have 42 - 64 + 1112 = 0 Demonstrate the Cayley-Hamilton Theorem for the matrix A given below. 03 1 A = -1 5 1 0 0 -1 STEP 1: Find and expand the characteristic equation. STEP 2: Compute the required powers of...
Hint: Apply the rank-nullity theorem to the linear map Pn → Rn+1 that sends p ?→...
Hint: Apply the rank-nullity theorem to the linear map Pn → Rn+1
that sends p ?→
(p(x0), . . . , p(xn)). Then use the fact that if polynomial of
degree ≤ n has n + 1 distinct roots, then it is the zero
polynomial.
(3 points) Application: polynomial interpolation. Let (20; yo), ..., (In; Yn) be n +1 points R2 with distinct x-coordinates. Show that there exists a unique polynomial p(t) of degree <n such that p(xi) = yi...
I know theorem states that rank T = Mdb(T). but what is next? Thank you very...
I know theorem states that rank T = Mdb(T). but what
is next?
Thank you very much
2. (8 marks) Let dim V = dim W = n. Let T:V → W be a linear transformation of rank k. Show that there are ordered bases B of V and D of W such that MdB(T) is a matrix that contains exactly k entries that are ls (with the rest being Os).
Anton Chapter 4, Section 4.8, Supplementary Question 01 Find the rank and nullity of the matrix; then verify that the values obtained satisfy Formula (4) in the Dimension Theorem. [1 4 6 5 8] 3 -4 2 -1 -40 1-1 0 -2 -1 8 [ 4 7 15 11 -4 A = 1 Click here to enter or edit your answer rank(A) = Click here to enter or edit your answer nullity(A) = Click here to enter or edit your...
Linear Algebra 1. One of the most important applications of linear algebra to electronics is to analyze electronic circuits that cannot be described using the rules for resistors in series or parallel such as the one shown figure in below. The goal is to calculate the current flowing in each branch of the circuit or to calculate the voltage at each node of the circuit. This circuit is called Loop Current Analysis of Electric Circuits. In this circuit and number...
About linear algebra,matrix;
2. (a) Use Octave as a Calculator to answer this question. Suppose that A and B are two 8 x 9 matrices. The (i.j)-entry of the matrix B is given by i *j -1. The (i. j)-entry of the matrix A equals 0 if i + j is divisible by 5 and equals the (i,j)-entry of the matrix B otherwise. i. What are the rank and nullity of matrices A and B? ii. Is vector u 9,64,-71,...
CHALLENGE ACTIVITY 5.5.1: Rank and nullity of a linear transformation. Jump to level 1 1 2 Let T:U + V be a linear transformation. Use the rank-nullity theorem to complete the information in the table below. > -D- D - D + 3 4 P5 6 Pg Ex: 5 Pn Ex: n+2 U dim(U) rank(T) nullity(T) 4 Ex: 5 6 Ex: n+2 7 Ex: 5 2. Check Next Feedback?
Advanced Linear Algebra (bonus problem)
1. (This question guides you through a different proof of part of the Decomposition Theorem. So you are not allowed to use the Decomposition Theorem when answering this question.) Let F be a field and V an n-dimensional F-vector space for n > I. Let θ E End(V) be a linear transformation and α E F an eigenvalue of. Recall that the generalised α-eigenspace of θ is a) Suppose that 0 υ Ε να and...
The Cayley-Hamilton Theorem states that a matrix satisfies its characteristic equation. For example, the characteristic equation of the matrix shown below is as follows. 1-3 A = 12 - 61 + 11 = 0 and by the theorem you have A2 - 64 + 1112 = 0 2 5 Demonstrate the Cayley-Hamilton Theorem for the matrix A given below. 0 5 -1 -1 3 1 0 0 1 STEP 1: Find and expand the characteristic equation. STEP 2: Compute the...
The Cayley-Hamilton Theorem states that a matrix satisfies its characteristic equation. For example, the characteristic equation of the matrix shown below is as follows. --1:: 22 - 61 + 11 = 0 and by the theorem you have 42 - 64 + 1112 = 0 Demonstrate the Cayley-Hamilton Theorem for the matrix A given below. 03 1 A = -1 5 1 0 0 -1 STEP 1: Find and expand the characteristic equation. STEP 2: Compute the required powers of...
Hint: Apply the rank-nullity theorem to the linear map Pn → Rn+1
that sends p ?→
(p(x0), . . . , p(xn)). Then use the fact that if polynomial of
degree ≤ n has n + 1 distinct roots, then it is the zero
polynomial.
(3 points) Application: polynomial interpolation. Let (20; yo), ..., (In; Yn) be n +1 points R2 with distinct x-coordinates. Show that there exists a unique polynomial p(t) of degree <n such that p(xi) = yi...
I know theorem states that rank T = Mdb(T). but what
is next?
Thank you very much
2. (8 marks) Let dim V = dim W = n. Let T:V → W be a linear transformation of rank k. Show that there are ordered bases B of V and D of W such that MdB(T) is a matrix that contains exactly k entries that are ls (with the rest being Os).
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