Question

Is the set of all x, y, z such x+ 3y + 2z = 0 a...

Is the set of all x, y, z such x+ 3y + 2z = 0 a subspace of R^3 ? If so find a basis for the space.

Homework Answers

Answer #1

Let X1 = (x1,y1,z1)T and X2 = (x2,y2,z2)T be 2 arbitrary vectors in V (say), the given subset of R3 and let k be an arbitrary scalar. Then x1+ 3y1 + 2z1 = 0 and x2+ 3y2 + 2z2 = 0. Further, X1+X2 = (x1,y1,z1)T +(x2,y2,z2)T = (x1+x2,y1+y2,z1+z2)T and (x1+x2)+ 3(y1 +y2)+ 2(z1+z2) = (x1+ 3y1 + 2z1)+( x2+ 3y2 + 2z2)=0+0 = 0. This implies that X1+X2 ∈ V so that V is closed under vector addition. Also kX1 = k(x1,y1,z1)T = (kx1,ky1,kz1)T and kx1+ 3ky1 + 2kz1 =k(x1+ 3y1 + 2z1 ) =k.0 = 0. This implies that kX1 ∈ V so that V is closed under scalar multiplication. Also, since 0+3.0+2.0 = 0, hence the zero vector (0,0,0)T ∈ V so that V is a vector space, and , therefore, a subspace of R3.

Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
Let W = {(x, y, z, w) ∈ R 4 | x − z = 0...
Let W = {(x, y, z, w) ∈ R 4 | x − z = 0 and y + 2z = 0} (a) Find a basis for W. (b) Apply the Gram-Schmidt algorithm to find an orthogonal basis for the subspace (2) U = Span{(1, 0, 1, 0),(1, 1, 0, 0),(0, 1, 0, 1)}.
1. Let W be the set of all [x y z}^t in R^3 such that xyz...
1. Let W be the set of all [x y z}^t in R^3 such that xyz = 0. Is W a subspace of R^3? 2. Let C^0 (R) denote the space of all continuous real-valued functions f(x) of x in R. Let W be the set of all continuous functions f(x) such that f(1) = 0. Is W a subspace of C^0(R)?
L(x,y,z)=(x+y+2z, x-z, 2x+3y-9z) f={(1,2,3) (-1,1,7), (3,0,5)} determine change of basis matrices
L(x,y,z)=(x+y+2z, x-z, 2x+3y-9z) f={(1,2,3) (-1,1,7), (3,0,5)} determine change of basis matrices
#7 Solve for x, y and z 6x+3y+z=-27 x-3y+2z+10 17x-2y+3z=-65
#7 Solve for x, y and z 6x+3y+z=-27 x-3y+2z+10 17x-2y+3z=-65
Consider the subspace S = {[x, y, 2x + 3y] | x, y ∈ R} of...
Consider the subspace S = {[x, y, 2x + 3y] | x, y ∈ R} of R 3 . (a) Find a basis of S and dim (S). (b) Extend the basis of S in (a) to a basis of R 3 .
1)T F: All (x, y, z) ∈ R 3 with x = y + z is...
1)T F: All (x, y, z) ∈ R 3 with x = y + z is a subspace of R 3 9 2) T F: All (x, y, z) ∈ R 3 with x + z = 2018 is a subspace of R 3 3) T F: All 2 × 2 symmetric matrices is a subspace of M22. (Here M22 is the vector space of all 2 × 2 matrices.) 4) T F: All polynomials of degree exactly 3 is...
Prove that the set S = {(x, y, z) ∈ R 3 : x + y...
Prove that the set S = {(x, y, z) ∈ R 3 : x + y + z = b}. is a subspace of R 3 if and only if b = 0.
Solve the system of equations using an inverse matrix -4x-2y+z= 6 -x-y-2z= -3 2x+3y-z= -4 Choose...
Solve the system of equations using an inverse matrix -4x-2y+z= 6 -x-y-2z= -3 2x+3y-z= -4 Choose one: a. (-1, 0, -2) b. (1, 0, -2) c. (1, 0, 2) d. (-1, 0, 2)
Use Gaussian elimination with backward substitution to solve the system of linear equations. x+y-z=-4 -x-4y+4z=1 -4x-3y+2z=15...
Use Gaussian elimination with backward substitution to solve the system of linear equations. x+y-z=-4 -x-4y+4z=1 -4x-3y+2z=15 What is the solution set?
Evaluate S (9x + y − 2z) dS. S: z = x + y/2 ,    0 ≤...
Evaluate S (9x + y − 2z) dS. S: z = x + y/2 ,    0 ≤ x ≤ 4,    0 ≤ y ≤ 3
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT