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Chapter 1: Problem 6

Prove that $\operatorname{tr}(a A+b B)=a \operatorname{tr}(A)+b \operatorname{tr}(B)\( for any \)A, B \in \mathrm{M}_{n \times n}(F) .$

Short Answer

Expert verified
To prove that \(\operatorname{tr}(aA + bB) = a \operatorname{tr}(A) + b \operatorname{tr}(B)\), we calculate the trace of (aA + bB) and the value of a * tr(A) + b * tr(B) and compare them: \(\operatorname{tr}(aA + bB) = \sum_{i=1}^{n} (aA_{ii} + bB_{ii})\) \(a \operatorname{tr}(A) + b \operatorname{tr}(B) = a \sum_{i=1}^{n} A_{ii} + b \sum_{i=1}^{n} B_{ii}\) As both expressions are equal, we can conclude that \(\operatorname{tr}(aA + bB) = a \operatorname{tr}(A) + b \operatorname{tr}(B)\).

Step by step solution

01

Calculate the trace of (aA + bB)

First, compute the sum of aA and bB. To do this, multiply each element of matrix A by a, and each element of matrix B by b. Then, add the corresponding elements of the two matrices. For a general element i,j, we have: \((aA + bB)_{ij} = aA_{ij} + bB_{ij}\) The trace of a matrix is the sum of its diagonal elements. Thus, we compute the trace of (aA + bB) as follows: \(\operatorname{tr}(aA + bB) = \sum_{i=1}^{n} (aA + bB)_{ii} = \sum_{i=1}^{n} (aA_{ii} + bB_{ii})\)
02

Calculate a * tr(A) + b * tr(B)

Next, we compute a times the trace of A and b times the trace of B. \(a \operatorname{tr}(A) = a \sum_{i=1}^{n} A_{ii}\) \(b \operatorname{tr}(B) = b \sum_{i=1}^{n} B_{ii}\) Now, sum these two values: \(a \operatorname{tr}(A) + b \operatorname{tr}(B) = a \sum_{i=1}^{n} A_{ii} + b \sum_{i=1}^{n} B_{ii}\)
03

Compare the two values

Now that we have computed the trace of (aA + bB) and the value of a * tr(A) + b * tr(B), we can compare them: \(\operatorname{tr}(aA + bB) = \sum_{i=1}^{n} (aA_{ii} + bB_{ii})\) \(a \operatorname{tr}(A) + b \operatorname{tr}(B) = a \sum_{i=1}^{n} A_{ii} + b \sum_{i=1}^{n} B_{ii}\) Since the two expressions have the same form and are equal, we can conclude that \(\operatorname{tr}(aA + bB) = a \operatorname{tr}(A) + b \operatorname{tr}(B)\).

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Most popular questions from this chapter

Determine whether the vectors emanating from the origin and terminating at the following pairs of points are parallel.(a) \((3,1,2)\) and \((6,4,2)\) (b) \((-3,1,7)\) and \((9,-3,-21)\) (c) \((5,-6,7)\) and \((-5,6,-7)\) (d) \((2,0,-5)\) and \((5,0,-2)\)

Exercises \(29-34\) require knowledge of the sum and direct sum of subspaces, as defined in the exercises of Section 1.3. (a) Prove that if \(W_{1}\) and \(W_{2}\) are finite-dimensional subspaces of a vector space \(V\), then the subspace \(W_{1}+W_{2}\) is finite-dimensional, and $\operatorname{dim}\left(\mathrm{W}_{1}+\mathrm{W}_{2}\right)=\operatorname{dim}\left(\mathrm{W}_{1}\right)+\operatorname{dim}\left(\mathrm{W}_{2}\right)-\operatorname{dim}\left(\mathrm{W}_{1} \cap \mathrm{W}_{2}\right)\(. Hint: Start with a basis \)\left\\{u_{1}, u_{2}, \ldots, u_{k}\right\\}\( for \)\mathrm{W}_{1} \cap \mathrm{W}_{2}$ and extend this set to a basis $\left\\{u_{1}, u_{2}, \ldots, u_{k}, v_{1}, v_{2}, \ldots, v_{m}\right\\}\( for \)\mathrm{W}_{1}\( and to a basis \)\left\\{u_{1}, u_{2}, \ldots, u_{k}, w_{1}, w_{2}, \ldots, w_{p}\right\\}$ for \(\mathrm{W}_{2}\). (b) Let \(W_{1}\) and \(W_{2}\) be finite-dimensional subspaces of a vector space \(\mathrm{V}\), and let \(\mathrm{V}=\mathrm{W}_{1}+\mathrm{W}_{2}\). Deduce that \(\mathrm{V}\) is the direct sum of \(\mathrm{W}_{1}\) and \(W_{2}\) if and only if \(\operatorname{dim}(V)=\operatorname{dim}\left(W_{1}\right)+\operatorname{dim}\left(W_{2}\right)\).

Let \(u\) and \(v\) be distinct vectors of a vector space \(\mathrm{V}\). Show that if \(\\{u, v\\}\) is a basis for \(\mathrm{V}\) and \(a\) and \(b\) are nonzero scalars, then both \(\\{u+v, a u\\}\) and \(\\{a u, b v\\}\) are also bases for \(\mathrm{V}\).

Show that the midpoint of the line segment joining the points \((a, b)\) and \((c, d)\) is \(((a+c) / 2,(b+d) / 2)\).

Let \(u, v\), and \(w\) be distinct vectors of a vector space \(\mathrm{V}\). Show that if \(\\{u, v, w\\}\) is a basis for \(\mathrm{V}\), then $\\{u+v+w, v+w, w\\}\( is also a basis for \)\mathrm{V}$.
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