Now we describe w 1 w 4 let p a 1 a 2 a 3 be a point

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Now we describe W 1 W 4 . Let P = ( a 1 , a 2 , a 3 ) be a point inside this intersection. Since P W 1 we can (just as above) write P = (3 a 2 , a 2 , - a 2 ). Since P W 4 we must have a 1 - 4 a 2 - a 3 = 0. Replacing a 1 by 3 a 2 and a 3 by - a 2 this equality becomes. 0 = a 1 - 4 a 2 - a 3 = (3 a 2 ) - 4 a 2 - ( - a 2 ) = 0 .
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4 HOMEWORK ASSIGNMENT 1 SOLUTIONS So what this says is that a point in W 1 already satisfies the equation defining W 4 . There- fore, W 1 W 4 = W 1 , which is again a subspace of R 3 (being closed under addition, scalar multiplication, and containing ~ 0). Now we describe W 3 W 4 . First notice that W 3 and W 4 are planes, both passing through the origin, and hence they cannot be parallel. Therefore their intersection is a line since they are not the same plane. Any point in this intersection has to satisfy the two equations a 1 - 4 a 2 - a 3 = 0 and 2 a 1 - 7 a 2 + a 3 = 0. From the first equation, we can solve for a 1 to get a 1 = 4 a 2 + a 3 . Plugging this into the second equation, we have 2(4 a 2 + a 3 ) - 7 a 2 + a 3 = 0. Solving for a 2 we get a 2 = - 3 a 3 . Using this we also find a 1 = 4 a 2 + a 3 = 4( - 3 a 3 ) + a 3 = - 11 a 3 . So W 3 W 4 = { ( a 1 , a 2 , a 3 ) R 3 : a 1 = - 11 a 3 , a 2 = - 3 a 3 } , which is indeed a line like we said above. Since both W 3 and W 4 are subspaces of R 3 , there intersection is also, by Theorem 1.4. § 1.3 # 11 The answer is “no.” This is because W is not closed under addition. For example, f ( x ) = x n + 1 , g ( x ) = - x n W but f ( x ) + g ( x ) = 1 / W . § 1.4 # 10 Let M 1 , M 2 , and M 3 be the matrices as defined in the problem. Let M = a b c d be a symmetric matrix, for some a, b, c, d F . Then we must have b = c , so M = a b b d . But then M = aM 1 + dM 2 + bM 3 . Hence, every symmetric matrix is a linear combination of M 1 , M 2 , M 3 . This show that span ( { M 1 , M 2 , M 3 } ) contains the set of symmetric matrices. But clearly, any linear com- bination of M 1 , M 2 , M 3 is symmetric, so span ( { M 1 , M 2 , M 3 } ) equals the set of symmetric matrices.
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