Show that W = {(x1, x2): x1 ≥ 0 and x2 ≥ 0}, with the standard operations, is not a subspace of R2.

Q: 3. Let W = M2x2(C) with zero determinant. Show that W is not a subspace of V = M2x2(C).

A:

Q: Prove that every line through the origin in R3 is a subspace of R3.

A: To prove that every line through the origin in R3 is a subspace of R3.

Q: Suppose that U and V are both subspaces of R . Prove that their intersection U ∩ V = {v : v ∈ U…

A:

Q: Consider the subspace V of R4 given by 1 3 1 V = span || -1 1 1

A: Solution:-

Q: 3. Determine whether X1 W E R? : x² + x2 = | X2 is a subspace of R².

A:

Q: 2. Find the closest point to y in the subspace W spanned by vị and v2. 3 1 -2 y V2 13 3

A: To find  The closest point y in the subspace W spanned by v1 and v2.

Q: Verify that W is a subspace of V. In each case, assume that V has the standard operations.W = {(x1,…

A:

Q: Determine whether H is a subspace of M22. Give reasons! -X IX H={A€ M22 : A = 0 2x

A: Set of all 2*2 matrices denoted by M22 forms a vector space. We have to determine whether H is a…

Q: Let Z E R³ | y Prove that Z is a subspace of R³. = Z

A:

Q: Give an example of a non-zero subspace of R^4 which has dimension > 1 and does not contain any of…

A: Consider the example f(x) = {2 + x + x2}, this given polynomial has dimension 3 with the basis {1,…

Q: A subspace always contains the zero vector.

A: True Because of subspace is a nonempty set.

Q: Determine: If W = the set of all vectors in R2 whose second component is the square of the first. is…

A:

Q: can a subspace of R^n have a dimension less than n.

A: Yes , it can be .

Q: Let vị 2,v2 8,and w Determine if w is in the subspace of R generated by v1 and v2.

A:

Q: Find a basis for the subspace of R3 spanned by

A: In this question, we find the basis for the span of subspace S. S={(-1,2,5),(3,03), (5,1,8)}

Q: Prove that W1 = {(a,b) E R² : 4a – b= 0} is a subspace of R2.

A:

Q: 1. Determine whether w - {[:]•r:a-4-0} X1 E R? : : x1 – 23 = X2 is a subspace of R?.

A:

Q: Produce an example of an element of Co (R, R) which is not an element of the following vector…

A: Correct answer is f(x)= Sin(x) which is infinitely many times continuously differentiable but it is…

Q: 3 Find a basis of the subspace of R' defined by the equation -7x1 – 5x2 – 4x3 = 0.

A:

Q: What is the dimension of the following subspace U of R2x2 ? 3 U = span({ }) -2 -2

A:

Q: b) Let W = { 0) * x, y e R}. Show that W is not a subspaces of M2x2. %3D

A:

Q: Let-{ », :x, y, a, b e R and x - a = y – b}.Is W a subspace?

A: We will use the one step test for vector subspace to check if W is a subspace or not. We know that…

Q: Verify that W is a subspace of V. In each case, assume that V has the standard operations.{(x, y, 4x…

A:

Q: Verify that V5 is a subspace of RS

A: Please check step 2 for solution.!

Q: Suppose that X = ER' :z + y + z = 0 and ² + y² = z² Determine (with justification) whether X is a…

A: Given  X=xyz∈ℝ3:x+y+z=0 and x2+y2=z2

Q: Show that W = {a + bx² € P₂|a, b = R} is a subspace of P2.

A: The solution is given as

Q: snip

A: We have vector space of all real numbers over real numbers And S is subspace of vector space given…

Q: Let W be the set of all 3 x 3 matrices whose trace is zero. Show that S is a subspace of M33.

A: Let W be the set of all 3×3 matrices whose trace is zero. Show that W is a subspace of M33.

Q: Let W be a subset of the vectors in R where W = X2 X4 -x3 = x, - x, }. Show that W is a X3 3. X4…

A: We will use the formal definition of a subspace of a vector space to prove that, W is a subspace of…

Q: Prove that if S1 and S2 are orthogonal subspacesof Rn, then their intersection consists of only the…

A:

Q: What is the dimension of w = {A € M2x2 (R) : A"| =ö} ĄT -2 1 2 This is not a subspace

A: W=AEM2×2(R) : AT1-2=→0  Let A=abcd, a,b,c,dER     AT=acbd     AT1-2=→0  ⇒acbd 1-2 =00 ⇒a-2cb-2d=00…

Q: 3 Find a basis of the subspace of R’ defined by the equation -5x1 – 7x2 – 2x3 = 0.

A: Consider the given equation:

Q: Find the dimension of the subspace of all vectors in R° whose second and fourth entries are equal.…

A:

Q: i) Determine whether the set W = {(x1, x2, x1x2) : xị and r2 are real numbers} is a subspace of R³…

A: Introduction: We prove the given set to be valid as per the standard conditions:

Q: Give a counter example to show that W is not subspace of R³. W=set of (x, y, z) where x + y = 1 6. ,

A:

Q: Prove that the the solution to the equation 2x – y+z= 0 is a subspace of R³.

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Q: In C[−π, π], find the dimension of the subspace spanned by 1, cos 2x, cos2 x.

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Q: (b) Define subspace and Verify whether is subspace v = {(x,y,2):x,y,2ER² and x-2y+3z = 5}

A:

Q: Suppose that X = eR' : 2 + y + z = 0 and r²+y² = z² } . Determine (with justification) whether X is…

A: We have to check whether the set X is a subspace of the vector space ℝ3. We have to give a…

Q: snip

A: We can represent the given set again as: W=xyz:x=3y+1 and z=-2y Let  w1=x1y1z1∈WThen,x1=3y1+1 and…

Q: Let A be an n x n matrix. Verify that V = {x e R" : Ax = 3x} is a subspace of R".

A:

Q: The set of all points in R3 satisfying x + y - z = 0 is a subspace. Note that the set of point…

A: It is known that any subspace of ℝ3 should contain an additivities identity element 0, 0, 0.

Q: Find the dimension of the subspace Hof R2 spanned by

A:

Q: determine the dimension of the subspace of R3 spanned by the given vectors.

A: Determine the dimension of the subspace of ℝ3 spanned by the given vectors.

Q: Determine the dimension of the subspace H, where H = -6

A:

Q: H is not a subspace of R.

A:

Q: Determine: if W = the set of all vectors in R^2 whose second component is the square of the first.…

A:

Q: If V is a 13 dimensional subspace of R20 then the dimension of V- must be

A: Definition of Orthogonal Complements Let V be a subspace of ℝn. Its orthogonal complement is the…

Q: Find the closest point to y in the subspace W spanned by v1 and v2 2 1 y = -2 -3] 1 2 а) b) 2 c) 1…

A:

Q: Find the orthogonal compliment of the subspace W of 2x2 upper triangular matrices in the vector…

A: Using standard inner product in the space of matrices.

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.