Linear Algebra 6. (6 points) 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: . of with c some constant, is equal to a new vector where every element in cu is equal to the corresponding element in f multiplied by c. For example, 2 2= . vi + us is equal to a new vector with elements equal to the elementwise addition of Up and 12. For example, + - [.2] The above properties form our definition for a linear combination of vectors. of is a linear combination of oh and op if on = au, + 6oz, where a and & are some constants. Oftentimes, we stack column vectors to form a matrix. Define the 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. 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. For each part below, you will be presented with a set of vectors, and a matrix consisting of those vectors stacked in columns. State the rank of the matrix, and whether or not the matrix is full rank. If the matrix is not full rank, state a linear relationship among the vectors for example: 01 = 02.