Prove that W1={ (al, a2, ..., an)e Fn: al+a2+ +an=0} is a subspace of Fn, but W2= { (al, a2, ..., an)e Fn: al+a2+ +an=1}is not.

Q: Let S be the set of symmetric 3 × 3 matrices. Determine whether S is a vector subspace of M3×3.

A: The answer is given below:

Q: Let M, N be closed linear subspaces of a Hilbert space H. If M 1 N, prove that M + N is closed.

A:

Q: 4) Show that the subspaces of R? are precisely {0}, R2, and all lines in R? through the origin.

A: A subspace of a vector space is itself a vector space that is contained in another vector space.

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: Let W be the set of all 2 × 2 matrices A = " 2 га b] such that Az = 0, where z = . Is W subspace of…

A: A set W of vector space V over field F is said to be subspace of vector space V if W is itself a…

Q: Show that W is a subspace of R¹.

A:

Q: 4.) Determine the dimension of thr subspace of M21 consisting of vectors of the form [a – 2b] a + b…

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Q: Q4(a) Define the null space of an m x n matrix A and prove that it is a subspace of R"

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Q: Determine whether the set of all matrices of the form (:: ;). a bc 1o), where a+e 0, and b +d+f 0,…

A: Given: abcdef , a+c=0, b+d+f=0

Q: let S be the collection of vector [x] (3x1 matrix) in R3 that satisfy the given…

A:

Q: b] Let W be the set of all 2 × 2 matrices A = such that Az = 0, where z = ||. Is W subspace of M22?…

A: A set W of vector space V over field F is said to be subspace of vector space V if W is itself a…

Q: Let W be the set of all 2 × 2 matrices A = " d. such that Az = 0, where z = |. Is W subspace of M22?…

A: Use the properties of a subspace or definition of subspace. Assume that the condition is true and…

Q: Let H be the set of all vectors of the form (a – 3b, b – a, a, b) where a and b are arbitrary…

A:

Q: Z. Suppose A and B are m x n matrices. Prove that C(A) and C(B) are orthogonal subspaces of R" if…

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Q: Let W {(a,b, c, d)| 5a - c = 0 } be a subspace of R*. Then a basis for W is

A:

Q: Prove that the set of all n x n matrices A such that AT = -A is a subspace of Mn,n.

A: We have to show that set of all skew symmetric matrix form subspace of Mnn

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

A:

Q: Find a basis for the following subspace of IR°. P = {¤ E R° | x1= x3 = x5, x2 + x4 = 0} What is the…

A: Explanation of the answer is as follows

Q: Let S=v1,v2,v3,v4, where v1= (1,1,2,1), v2= (1,0,−3,1),v3= (0,1,1,2),v4= (0,0,1,1), and v5=…

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Q: Let S be the subspace of R4 spanned by x1 = (1, 0,−2, 1)T and x2 = (0, 1, 3,−2)T . Find a basis for…

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Q: 4. Use the definition of a subspace to prove that the set V:= :, y, z E R, y+ : 0 is a subspace of…

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Q: Prove that W1 = {(a1, a2, . . . , an) ∈ Fn: a1 + a2 + · · · + an= 0} is a subspace of Fn , but W2 =…

A: Prove that W1 = {(a1, a2, . . . , an) ∈ Fn: a1 + a2 + · · · + an= 0} is a subspace of Fn , but W2 =…

Q: Let W {(a, b, c, d)| a – 5 b = 0 } be a subspace of R*. Then a basis for W is

A: The given subspace of R4 is W={ ( a , b  ,c , d ) | a - 5 b = 0 } Let x=( a, b , c , d ) be in W…

Q: 3.8 Using the Gram-Schmidt method, turn the basis B = (b1, b2) of a two- dimensional subspace U C R…

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Q: Prove that if S ⊆ R is a vector subspace of R, then either S = {0} or S = R.

A: Here, if s={0} then we are done.  But suppose that S⊂ℝ and S≠{0} is a vector subspace of ℝ to prove…

Q: at | f = a is a subspace of RI0,1] if and

A: Two step test

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

A:

Q: Let w = ( { (x.y,z) ER3: x+y-z 0) be a subspace. A basis of w+ is [A].

A: Solution

Q: If S, is a subspace of R* of dimension 3, then there cannot exist a subspace S, of R* such that Si c…

A: S1, S2 are subspaces of R4 such that S1⊂S2⊂R4 with S1≠ S2 and S2≠R4.

Q: 2. Is the set of vectors of the form a b a+2b a subspace of R³? Show your proof.

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Q: The set {(x,3): x is in R)} is a subspace of R2

A: The given set is x,3: x is in R. It is known that any subspace should contain identity element.

Q: Let A be an mxn matrix. Prove that every vector x in R" can be written in the form x = p+u, where p…

A: Let A be an m×n matrix.Prove: for evert vector x∈ℝn can be written in the form x=p+u, where p is in…

Q: Let H be the set of all vectors of the form 3s . Find a vector v in R3 such that H = Span {v}. Why…

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Q: Let P, be the set of all polynomial functions in P(R) of the form ·+ anx", for some fixed positive…

A:

Q: 2. The set P3 of all polynomials of degree ≤ 3 with normal polynomial operations is a vector space.…

A: Check which of the following are subspaces of P3.

Q: Given a basis B = {V1, V2,, Vn} of Subspace S, there is a unique way to express any ... vE S as a s…

A: We will first write 2 expressions of v as a linear combination of basis vectors in B. Then, we will…

Q: 3). Determine whether the set of all matrices of the form a b c de f where a +c= 0, and b+d+ f = 0,…

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Q: Let Mnn be a vector space of n x n matrices. Which of the following is not a subspace of Mnn?

A: The vector space of nxn matrix is Mnn.

Q: Show that the set of all polynomials of degree at most 3 forms a subspace of Pn.

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Q: a) Prove that an operator P is the pr osed linear subspace M of a Hilbert

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Q: Let W be an (n – 1)-dimensional subspace of V. Show that V has a basis B satisfying BnW = Ø. |

A: This question is related to vector space . Solution of this question is

Q: Which of the following sets are vector subspaces of R3 and why? i. S1 ={x in R3: x1 + x2 = 0, x1 +…

A: To determine the set S is a subspace if w1 , w2∈S then c1w1+c2w2∈S (i) Consider…

Q: Let p(x) be a polynomial in P4. Let S be the set of all polynomials from P4 with p(0) = 0. Is Sa…

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Q: Is the set M={1/n | n ε z+} compact as a subspace of R?

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Q: Find a basis for orthonormal complement W- for a set W, a subspace in R* W = [2x, + x, + X + 2x, = 0…

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Q: Let A be an mxn matrix. Prove that W = {x ER" | Ax = 0} is a subspace of R", where W is the solution…

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Q: Show that W={(x1,x2,x3,x4) : x4-x3=x2-x1} is a subspace of R^4  spanned by vectors…

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Q: i) Let V = M2x2(R) and %3D W; = {(: :) ev:a.hceR} and W2 = {(". :) ev :a,be R Prove that W1 and W2…

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Correlation

Correlation defines a relationship between two independent variables. It tells the degree to which variables move in relation to each other. When two sets of data are related to each other, there is a correlation between them.

Linear Correlation

A correlation is used to determine the relationships between numerical and categorical variables. In other words, it is an indicator of how things are connected to one another. The correlation analysis is the study of how variables are related. 

Regression Analysis

Regression analysis is a statistical method in which it estimates the relationship between a dependent variable and one or more independent variable. In simple terms dependent variable is called as outcome variable and independent variable is called as predictors. Regression analysis is one of the methods to find the trends in data. The independent variable used in Regression analysis is named Predictor variable. It offers data of an associated dependent variable regarding a particular outcome.