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ints) A common representation of data uses matrices and vectors, so it is helpful to familiarize ourselves with linear algebra notation, as well as some simple operations. Define a vector ♬ to be a column vector. Then, the following properties hold: • cu with c some constant, is equal to a new vector where every element in cv is equal to the corresponding element in & multiplied by c. For example, 2 2 = ● √₁ + √2 is equal to a new vector with elements equal to the elementwise addition of ₁ and 2. For example, 問 2+4-6 = The above properties form our definition for a linear combination of vectors. √3 is a linear combination of √₁ and √2 if √3 = a√₁ + b√2, where a and b are some constants. Oftentimes, we stack column vectors to form a matrix. Define the column rank of a matrix A to be equal to the maximal number of linearly independent columns in A. A set of columns is linearly independent if no column can be written as a linear combination of any other column(s) within the set. If all columns in a matrix are linearly independent, it means that the matrix is full column rank. For example, let A be a matrix with 4 columns. If three of these columns are linearly independent, but the fourth can be written as a linear combination of the other three, then rank(A) : 3. Alternatively, if all four columns of A were linearly independent, rank(A) = 4, and A would be full column rank. = For each of the following matrices, state the rank of the matrix and whether or not the matrix is full column rank. If the matrix is not full column rank, also give a linear relationship among the vectors—for example: √₁ = √2. (a) v₁ = 0 = A = "

ints) A common representation of data uses matrices and vectors, so it is helpful to familiarize ourselves with linear algebra notation, as well as some simple operations. Define a vector ♬ to be a column vector. Then, the following properties hold: • cu with c some constant, is equal to a new vector where every element in cv is equal to the corresponding element in & multiplied by c. For example, 2 2 = ● √₁ + √2 is equal to a new vector with elements equal to the elementwise addition of ₁ and 2. For example, 問 2+4-6 = The above properties form our definition for a linear combination of vectors. √3 is a linear combination of √₁ and √2 if √3 = a√₁ + b√2, where a and b are some constants. Oftentimes, we stack column vectors to form a matrix. Define the column rank of a matrix A to be equal to the maximal number of linearly independent columns in A. A set of columns is linearly independent if no column can be written as a linear combination of any other column(s) within the set. If all columns in a matrix are linearly independent, it means that the matrix is full column rank. For example, let A be a matrix with 4 columns. If three of these columns are linearly independent, but the fourth can be written as a linear combination of the other three, then rank(A) : 3. Alternatively, if all four columns of A were linearly independent, rank(A) = 4, and A would be full column rank. = For each of the following matrices, state the rank of the matrix and whether or not the matrix is full column rank. If the matrix is not full column rank, also give a linear relationship among the vectors—for example: √₁ = √2. (a) v₁ = 0 = A = "

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter5: Orthogonality
Section: Chapter Questions
Problem 2RQ
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ints) A common representation of data uses matrices and vectors, so it is helpful to familiarize ourselves with linear algebra notation, as well as some simple operations. Define a vector ♬ to be a column vector. Then, the following properties hold: • cu with c some constant, is equal to a new vector where every element in cv is equal to the corresponding element in & multiplied by c. For example, 2 2 = ● √₁ + √2 is equal to a new vector with elements equal to the elementwise addition of ₁ and 2. For example, 問 2+4-6 = The above properties form our definition for a linear combination of vectors. √3 is a linear combination of √₁ and √2 if √3 = a√₁ + b√2, where a and b are some constants. Oftentimes, we stack column vectors to form a matrix. Define the column rank of a matrix A to be equal to the maximal number of linearly independent columns in A. A set of columns is linearly independent if no column can be written as a linear combination of any other column(s) within the set. If all columns in a matrix are linearly independent, it means that the matrix is full column rank. For example, let A be a matrix with 4 columns. If three of these columns are linearly independent, but the fourth can be written as a linear combination of the other three, then rank(A) : 3. Alternatively, if all four columns of A were linearly independent, rank(A) = 4, and A would be full column rank. = For each of the following matrices, state the rank of the matrix and whether or not the matrix is full column rank. If the matrix is not full column rank, also give a linear relationship among the vectors—for example: √₁ = √2. (a) v₁ = 0 = A = "
Transcribed Image Text:ints) A common representation of data uses matrices and vectors, so it is helpful to familiarize ourselves with linear algebra notation, as well as some simple operations. Define a vector ♬ to be a column vector. Then, the following properties hold: • cu with c some constant, is equal to a new vector where every element in cv is equal to the corresponding element in & multiplied by c. For example, 2 2 = ● √₁ + √2 is equal to a new vector with elements equal to the elementwise addition of ₁ and 2. For example, 問 2+4-6 = The above properties form our definition for a linear combination of vectors. √3 is a linear combination of √₁ and √2 if √3 = a√₁ + b√2, where a and b are some constants. Oftentimes, we stack column vectors to form a matrix. Define the column rank of a matrix A to be equal to the maximal number of linearly independent columns in A. A set of columns is linearly independent if no column can be written as a linear combination of any other column(s) within the set. If all columns in a matrix are linearly independent, it means that the matrix is full column rank. For example, let A be a matrix with 4 columns. If three of these columns are linearly independent, but the fourth can be written as a linear combination of the other three, then rank(A) : 3. Alternatively, if all four columns of A were linearly independent, rank(A) = 4, and A would be full column rank. = For each of the following matrices, state the rank of the matrix and whether or not the matrix is full column rank. If the matrix is not full column rank, also give a linear relationship among the vectors—for example: √₁ = √2. (a) v₁ = 0 = A = "
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