T or F 1) Any N vectors spanning R^n are linearly independent 2)R5 has 7 linearly...

T or F

1) Any N vectors spanning R^n are linearly independent

2)R5 has 7 linearly independent vectors

3) If a set of vectors with n elements is linearly dependent, then a set with n - 1 elements is also linearly dependent

4) There exists a Linear Function T:R^n -> R^n such that the range and the kernel of T are equal.

5) If a vector space has a dimension of n, then a basis for the vector space will contain n vectors

6) L: R^6 -> R^7 is one-to-one, then the range of L has a dimension of 7

7) The number of leading terms in ref(A) is equal to the dimension of the row space of matrix A

8) If a set of 4 vectors is linearly independent, then if you remove 1 vector, the set will still be linearly independent

Expert Solution

Colby Messinger answered 3 months ago

Let p and q be two linearly independent vectors in R^n such that ||p||_2=1, ||q||_2=1 ....

Let p and q be two linearly independent vectors in R^n such that ||p||_2=1, ||q||_2=1 . Let A=pq^T+qp^T. determine the kernel, nullspace,rank and eigenvalue decomposition of A in terms of p and q.

Show that, in n-dimensional space, any n + 1 vectors are linearly dependent. HINT: Given n+1...

Show that, in n-dimensional space, any n + 1 vectors are linearly dependent. HINT: Given n+1 vectors, where each vector has n components, write out the equations that determine whether these vectors are linearly dependent or not. Show that these equations constitute a system of n linear homogeneous equations with n + 1 unknowns. What do you know about the possible solutions to such a system of equations?

Find the vectors T, N, and B at the given point. r(t) = < t2, (2/3)...

Find the vectors T, N, and B at the given point. r(t) = < t2, (2/3) (t3), t > , < 4, -16/3, -2>

1. Determine each of the following set of vectors is linearly independent or dependent. (a) S1...

1. Determine each of the following set of vectors is linearly independent or dependent. (a) S1 = {(1, 2, 3),(4, 5, 6),(6, 9, 12)}. (b) S2 = {(1, 2, 3, 4),(5, 6, 7, 8),(3, 2, 1, 0)}. (c) S3 = {(1, 2, 3, 4),(5, 6, 7, 8),(9, 10, 11, 12)}

Check whether the following families of functions of t are linearly independent or not (a) t^2...

Check whether the following families of functions of t are linearly independent or not (a) t^2 + 1, 2t, 4(t + 1)^2 (b) sin(t) cos(t), sin(2t) + cos(2t), cos(2t) (c) e^2t , e^-2t , 2e^t (d) 2e^t , 3 cosh(t), 13 sinh(t) (e) 1/((t^2)-1) , 1/(t + 1), 1/(t-1)

R= Ro(1/2)n n= number of half lifetimes= t/t1/2 (a)n= 2/2= 1 R= Ro(1/2)n R=3000(1/2)1 R= 1500...

R= Ro(1/2)n n= number of half lifetimes= t/t1/2 (a)n= 2/2= 1 R= Ro(1/2)n R=3000(1/2)1 R= 1500 counts/sec (b)n= 6/2= 3 R= Ro(1/2)n R=3000(1/2)3 R= 375 counts/sec (c) n= 10/2= 5 R= Ro(1/2)n R=3000(1/2)5 R= 93.75 counts/sec (d) n= 20/2= 10 R= Ro(1/2)n R=3000(1/2)10 R= 2.93 counts/sec What is the mean life of this nucleus? f. Suppose that the Geiger counter detects 10% of all the radioactive decays. What is the total number of radioactive nuclei at time t = 0?...

Let ?1=(1,0,1,0) ?2=(0,−1,1,−1) ?3=(1,1,1,1) be linearly independent vectors in ℝ4. a. Apply the Gram-Schmidt algorithm to...

Let ?1=(1,0,1,0) ?2=(0,−1,1,−1) ?3=(1,1,1,1) be linearly independent vectors in ℝ4. a. Apply the Gram-Schmidt algorithm to orthonormalise the vectors {?1,?2,?3} of vectors {?1,?2,?3}. b. Find a vector ?4 such that {?1,?2,?3,?4} is an orthonormal basis for ℝ4 (where ℝ4 is the Euclidean space, that is, the inner product is the dot product).

(1 point) For the given position vectors r(t)r(t) compute the unit tangent vector T(t)T(t) for the...

(1 point) For the given position vectors r(t)r(t) compute the unit tangent vector T(t)T(t) for the given value of tt . A) Let r(t)=〈cos5t,sin5t〉 Then T(π4)〈 B) Let r(t)=〈t^2,t^3〉 Then T(4)=〈 C) Let r(t)=e^(5t)i+e^(−4t)j+tk Then T(−5)=

proof: L t^(n+1)f(t)=(-1)^(n+1)(d^(n+1)/ds^(n+1))*F(s)

proof: L t^(n+1)*f(t)=(-1)^(n+1)*(d^(n+1)/ds^(n+1))*F(s)

For problems 1-4 the linear transformation T/; R^n - R^m is defined by T(v)=AV with A=[-1 -2...

For problems 1-4 the linear transformation T/; R^n - R^m is defined by T(v)=AV with A=[-1 -2 -1 -1 2 1 0 -4 4 6 1 15] 1. Find the dimensions of R^n and R^m (3 pts) 2. Find T(<-2, 1, 4, -1>) (5 pts) 3. Find the preimage of <-6, 12, 4> (8 pts) 4. Find the Ker(T) (8 pts)

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