Let U₁, U2, U3 – be the following subspaces of Rª U₁={a, b, c, d)=R4| ²b=c; a=d=0}; U2={(a, b, c, d)=R4| a - b + 3¹c + 4;d=0, -a +b - 3¹c - 4 d=0}; U3= {a, b, c, d)=R4| 2a − 3b − 2 c - d=0, a + 2b + 0c + .² d=0 }; and the dimension () of U₁. Sub-Task 1. Find a basis Sub-Task 2. Find a basis () and the dimension () of U₂. Sub-Task 3. Find a basis (^.¯ _ `) and the dimension (´¸ ¯`) of U3. Sub-Task 4. Whether Rª=U₁+U2, (provide a justification). (. ). Whether R¹=U₁U2, (provide a justification). (~ 1 Sub-Task 5. Whether Rª=U₁+U3, (provide a justification) ^ Whether Rª=U₁ÐU3, (provide a justification). (^´ ^^). Sub-Task 6. Whether Rª=U2+U3, (provide a justification). (u. (provide a justification). (¯¸). *). Whether R4=U₂ÐU3,

Q: Determine whether the set in P_2 is linearly independent.    S={x^2 , +1, 2 - 1, 2}

A:

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A: (a), (b) , (c), (d)  are subspaces .

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A: .

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A: We will check that which subset is not a subspace of R2

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A:

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A: image is attached with detailed explnation

Q: 19. Prove that the set of all vectors in R³that satisfy the equation x₁ = 3x2, is a subspace.

A: (.)  Let set  'S'  is given by ,        S=x1,x2,x3∈ℝ3  :  x1=3x2 or   S= x1,x2,x3∈ℝ3 :  x1-3x2=0…

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A:

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A: Let the subset W is given as W={(x1,x2,x3)| x1+x2+x3=0} Determine whether the given subset is…

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A: