Homework Answers
first
we find matrix representation of T with respect to the standard
basis.then T is invertible iff A is invertible.but here det A=0
(expanding along thirdd row of A) so A is not invertible hence T is
not invertible so inverse of T does not exist.
28. Show that there are 12 pairs of numbers (a1,az) with 0<aj < 4,0 <a2 <6...
28. Show that there are 12 pairs of numbers (a1,az) with 0<aj < 4,0 <a2 <6 so that x=a1 (mod 4) x = 22 (mod 6) has a solution.
1. Let az, az, az, a4 are vectors in R3. Suppose that az 3a1 – 2a3...
1. Let az, az, az, a4 are vectors in R3. Suppose that az 3a1 – 2a3 + 84. (a) Are aj, aj, az, a4 linearly independent? (b) Suppose that ai, az, a4 are linearly independent. What is the dimension of the span{a1, az, az, a4}? (c) Is the set of vectors aj, az, az, a4 form a basis of R3? Explain your reasoning. (d) Form a basis of R3 using a subset of ai, a2, a3, 24.
Let ai, a2 , аз, bị, b2P3 R. Define T : R3 R2 by Prove T...
Let ai, a2 , аз, bị, b2P3 R. Define T : R3 R2 by Prove T is a linear transformation.
(6) Let T R R² be defined by T (a, az) = (a, -a2, a., 29,...
(6) Let T R R² be defined by T (a, az) = (a, -a2, a., 29, +a). Let ß be the standard basis for 1R² and v= {(1,1,0, (0, 1, 1), (2, 3,3)} Compute [7]} .
QUESTION 4 Let T R3-P2 be defined by T(a, b, c) - (a + b + e) +(a+b)a2 (4.1) Show that T is a linear transformation (4.2) Fınd the matrix representation [T]s, B, of T relative to the basıs in R3 and...
QUESTION 4 Let T R3-P2 be defined by T(a, b, c) - (a + b + e) +(a+b)a2 (4.1) Show that T is a linear transformation (4.2) Fınd the matrix representation [T]s, B, of T relative to the basıs in R3 and the basis in P2, ordered from left to right Determine the range R(T of T Is T onto? In other words, is it true that R(T)P2 Let x, y E R3 Show that x-y ker(T) f and only...
Let R3 to R2 be a vector function. Then determine whether T is a linear transformation or not. Defined by T(a1,a2,a3) = (0,a3)
T: R3 to R 2 vector function.Is T a linear transformation or not defined by T(a1,a2, a3) = (0, a3 )
Consider the three 4-dimensional vectors aj = _21, 22 = 1 , a3 = 11 and...
Consider the three 4-dimensional vectors aj = _21, 22 = 1 , a3 = 11 and the matrix A = [a], 22, az). (a) Find rank A and null A. (b) The linear transformation TA : R3 → R4 is defined by T.(x) = Ax. Determine whether TA is injective or not. (c) Determine whether the vectors aj, a2, az are linearly independent or dependent.
1: We define the Vandermonde Determinant, denoted V(ai,a2,... ,an), as ai a...a-1 1 2 i a2 az...a-1 al,a2 ,an ) 2....
1: We define the Vandermonde Determinant, denoted V(ai,a2,... ,an), as ai a...a-1 1 2 i a2 az...a-1 al,a2 ,an ) 2. 1 an a an ...an-1 We will guide you through a proof by Mathematical Induction to show that V(a,aan) aj -ai f: Show that if we perform k Type 3 ccolumn operations by adding a multiple B, of col- umn i, where1,2,. ,k, to the last column, then the Vandermonde determinant of size (k 1) x (k 1) can...
Let V be the vector space of all sequences over R. Given (a1, a2, T,U V...
Let V be the vector space of all sequences over R. Given (a1, a2, T,U V V by ) e V, define : ) ...) = (0, a1, 0, a2, 0, a3, . . . ) Тај, а2, аз, ад, 0, аз, (a1, a3, a5,.) and U(a1, a2, a3, a4, (a) Find N(T) and N(U) (b) Explain why T is onto, but not 1-1 (c) Explain why U is 1-1, but not onto.
28. Show that there are 12 pairs of numbers (a1,az) with 0<aj < 4,0 <a2 <6 so that x=a1 (mod 4) x = 22 (mod 6) has a solution.
1. Let az, az, az, a4 are vectors in R3. Suppose that az 3a1 – 2a3 + 84. (a) Are aj, aj, az, a4 linearly independent? (b) Suppose that ai, az, a4 are linearly independent. What is the dimension of the span{a1, az, az, a4}? (c) Is the set of vectors aj, az, az, a4 form a basis of R3? Explain your reasoning. (d) Form a basis of R3 using a subset of ai, a2, a3, 24.
Let ai, a2 , аз, bị, b2P3 R. Define T : R3 R2 by Prove T is a linear transformation.
(6) Let T R R² be defined by T (a, az) = (a, -a2, a., 29, +a). Let ß be the standard basis for 1R² and v= {(1,1,0, (0, 1, 1), (2, 3,3)} Compute [7]} .
QUESTION 4 Let T R3-P2 be defined by T(a, b, c) - (a + b + e) +(a+b)a2 (4.1) Show that T is a linear transformation (4.2) Fınd the matrix representation [T]s, B, of T relative to the basıs in R3 and the basis in P2, ordered from left to right Determine the range R(T of T Is T onto? In other words, is it true that R(T)P2 Let x, y E R3 Show that x-y ker(T) f and only...
Consider the three 4-dimensional vectors aj = _21, 22 = 1 , a3 = 11 and the matrix A = [a], 22, az). (a) Find rank A and null A. (b) The linear transformation TA : R3 → R4 is defined by T.(x) = Ax. Determine whether TA is injective or not. (c) Determine whether the vectors aj, a2, az are linearly independent or dependent.
1: We define the Vandermonde Determinant, denoted V(ai,a2,... ,an), as ai a...a-1 1 2 i a2 az...a-1 al,a2 ,an ) 2. 1 an a an ...an-1 We will guide you through a proof by Mathematical Induction to show that V(a,aan) aj -ai f: Show that if we perform k Type 3 ccolumn operations by adding a multiple B, of col- umn i, where1,2,. ,k, to the last column, then the Vandermonde determinant of size (k 1) x (k 1) can...
Let V be the vector space of all sequences over R. Given (a1, a2, T,U V V by ) e V, define : ) ...) = (0, a1, 0, a2, 0, a3, . . . ) Тај, а2, аз, ад, 0, аз, (a1, a3, a5,.) and U(a1, a2, a3, a4, (a) Find N(T) and N(U) (b) Explain why T is onto, but not 1-1 (c) Explain why U is 1-1, but not onto.
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