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Chapter 1: Problem 17

Let \(M\) be a square upper triangular matrix (as defined on page 19 of Section 1.3) with nonzero diagonal entries. Prove that the columns of \(M\) are linearly independent.

Short Answer

Expert verified
The columns of a square upper triangular matrix with nonzero diagonal entries are linearly independent. To prove this, we set up a linear combination of the columns equal to the zero vector and solve the resulting system of linear equations. By doing so, we find that all coefficients in the linear combination must be zero, so no other linear combination of the columns gives the zero vector. This confirms that the columns are linearly independent.

Step by step solution

01

Understand the Key Concepts

Review the key concepts relevant to this problem: (a) A square matrix is a matrix with the same number of rows and columns. (b) An upper triangular matrix is a special kind of square matrix where all the entries below the diagonal are zero. (c) Columns of a matrix can be treated as vectors. The concept of linear independence among vectors is that no vector can be expressed as a linear combination of others.
02

Understand the Given Matrix Structure

The given matrix \( M \) is a square upper triangular matrix with nonzero diagonal elements. Let's say, for instance: \[ M = \begin{bmatrix} a & b & c \\ 0 & d & e \\ 0 & 0 & f \end{bmatrix} \] In this matrix, \( a, d, \) and \( f \) are the diagonal elements, which are given to be nonzero.
03

Formulate Conditions for Linear Independence

For columns of a matrix to be linearly independent, a linear combination of those columns must only be the zero vector if all coefficients are zero. So, for \( M \), we set a linear combination of its columns equal to the zero vector: \[ x_1 \begin{bmatrix} a \\ 0 \\ 0 \end{bmatrix} + x_2 \begin{bmatrix} b \\ d \\ 0 \end{bmatrix} + x_3 \begin{bmatrix} c \\ e \\ f \end{bmatrix} = 0 \]
04

Express in Terms of Equations

From the above, we can express three simultaneous equations: 1. \( a x_1 + b x_2 + c x_3 = 0 \) 2. \( d x_2 + e x_3 = 0 \) 3. \( f x_3 = 0 \)
05

Solve the System of Equations

Solve the equations one by one, starting from the last one. From equation (3), we have \( x_3 = 0 \) because \( f \) is non-zero. Substituting \( x_3 = 0 \) into equation (2), we get \( d x_2 = 0 \), which implies \( x_2 = 0 \) as \( d \) is also nonzero. Lastly, substituting these into equation (1), we get \( a x_1 = 0 \), and thus \( x_1 = 0 \) because \( a \) is nonzero.
06

Conclude From the Results

Since all of our coefficients \( x_1, x_2, x_3 \) are equal to zero, this confirms that our column vectors are linearly independent because no other linear combination of them gives the zero vector. Therefore, we can conclude that the columns of a square upper triangular matrix with nonzero diagonal entries are linearly independent.

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Most popular questions from this chapter

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 \(\operatorname{dim}(V)=\operatorname{dim}\left(W_{1}\right)+\operatorname{dim}\left(W_{2}\right)\).

The vectors \(u_{1}=(1,1,1,1), u_{2}=(0,1,1,1), u_{3}=(0,0,1,1)\), and \(u_{4}=(0,0,0,1)\) form a basis for \(F^{4}\). Find the unique representation of an arbitrary vector \(\left(a_{1}, a_{2}, a_{3}, a_{4}\right)\) in \(\mathrm{F}^{4}\) as a linear combination of \(u_{1}, u_{2}, u_{3}\), and \(u_{4}\).

Show that the set \(\left\\{1, x, x^{2}, \ldots, x^{n}\right\\}\) is linearly independent in \(\mathrm{P}_{n}(F)\).

Let \(W\) be a subspace of a (not necessarily finite-dimensional) vector space \(V\). Prove that any basis for \(W\) is a subset of a basis for \(V\).

Let \(C^{n}(R)\) denote the set of all real-valued functions defined on the real line that have a continuous \(n\)th derivative. Prove that \(C^{n}(R)\) is a subspace of \(\mathcal{F}(R, R)\).
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