Let H and K be subspaces of a vector space V. The intersection of H and K, writt
ID: 3006882 • Letter: L
Question
Let H and K be subspaces of a vector space V. The intersection of H and K, written H K, is the set of v in V that belong to both H and K. Show that H K is a subspace of V. Suppose V and W are both vector spaces. We can create a new collection of objects, which we call V W, consisting of symbols (u,w), where v is a "vector" in V and w is a "vector" in W. Declare addition on this collection of objects to be the operation where (v_1, w_1) + (v_2, w_2) = (v_1 + v_2, w_1 + w_2), and declare scalar multiplication to be the operation where c(v, w) = (cv, cw). This makes V W a vector space. Explain what the 0 "vector" in V W is and show it has the right properties. Label every + sign you write as the + operation for V, the + operation for W, or this newly declared + operation for V W. Let M_2 Times 2 be the vector space of all 2 Times 2 matrices, and define the function T : M_2 Times 2 rightarrow M_2 Times 2 by T (A) = A + A^T (where A^T Ls the matrix A flipped across the diagonal). Explain why T is a linear transformation. Show that the image of T consists of all matrices B with the property that B = B^T. Describe the kernel of T. Define a linear transformation L : P_2 rightarrow R^2 by T(p) = [p(1) P(2)]. Find a polynomial p that spans the kernel of L. Explain why every vector in R^2 is in the image of L.Explanation / Answer
Q.1.
As 0 H and 0 K, 0 (H K).
Let u and v are two vectors in H K which gives u, v H, and u, v K.
This implies that u + v H (as H is a subspace), and similarly for K.
Thus, we get u + v (H K).
Let c be a scalar and v H K. This means that v must be common to both H and K.
Since v H, cv H. That means H is closed under scalar multiplication.
Similarly, as K is a subspace, and v K, cv K. As, cv is common to both H and K, so cv H K.
Thus, it is proved that (H K) is a subspace of V .
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