Linear Combinations of Vectors Span Dimension and Linear

Linear Combinations of Vectors Span, Dimension and Linear Independence Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Linear Combinations and Span The span of a set of vectors is the collection of all possible linear combinations of those vectors. A basis for a set of vectors is a linearly independent subset that has the same span as the entire set. Linear Independence If there is some nonzero solution the set of vectors is linearly dependent. This implies some redundancy in the set of vectors. Prepared by Vince Zaccone i. e. we can write one of the vectors in terms of the others. For Campus Learning Assistance Services at UCSB

Linear Independence A set of vectors in ℝm is said to be linearly independent if the vector equation has only the trivial solution x 1=x 2=…=xn=0 If there is some nonzero solution the set of vectors is linearly dependent. This implies some redundancy in the set of vectors. i. e. we can write one of the vectors in terms of the others. When we put the vectors in a matrix and do row reduction, the number of pivots (or basic variables) corresponds to the dimension of the span of the set. If there any free variables, then the set is dependent. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Linear Combinations and Span Describe the span of each set. The span of a set of vectors is the collection of all possible linear combinations of those vectors. In this case there is only one vector in the set, so the span contains all the constant multiples of this vector. All of these vectors are on a line. Specifically the line y=x. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Linear Combinations and Span Describe the span of each set. The span of a set of vectors is the collection of all possible linear combinations of those vectors. This ends up being the same line as before, because the new vector is just a multiple of the original, so it adds nothing new to the span. Note that either vector alone could be a basis for set V 2. The span of V 2 is 1 -dimensional, so any vector on the line would be a basis. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Linear Combinations and Span Describe the span of each set. The span of a set of vectors is the collection of all possible linear combinations of those vectors. Let’s see a few examples of vectors in this span: Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Linear Combinations and Span Describe the span of each set. The span of a set of vectors is the collection of all possible linear combinations of those vectors. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Linear Combinations and Span Describe the span of each set. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Linear Independence Example: Are the following sets of vectors linearly independent? Describe the span of each set. V 1 spans ℝ 2. We will test this one with the definition. Try to solve this equation: This is equivalent to a system of 2 equations: Or an augmented matrix, with the vectors as columns: There is no nonzero solution, so the set is independent. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Linear Independence Example: Are the following sets of vectors linearly independent? Describe the span of each set. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Linear Independence Example: Are the following sets of vectors linearly independent? Describe the span of each set. V 2 spans ℝ 2. Notice that this set contains 3 vectors from ℝ 2, and we already know that the first 2 vectors are independent (from the previous set). V 2 must be a dependent set. The dimension of the vectors determines a maximum number of independent vectors that can be in a set. The span of this set is ℝ 2. You could say that the third vector doesn’t add anything new– it must be in the span of the others. Another way to see that the set is dependent is to write one of the vectors in terms of the other vectors in the set. This amounts to rearranging the equation Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Linear Independence Example: Are the following sets of vectors linearly independent? Describe the span of each set. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Linear Independence Example: Are the following sets of vectors linearly independent? Describe the span of each set. V 3 spans ℝ 3. Set up the augmented matrix for this one. It is already in RREF. Since there is only the zero solution, the set is independent. Note: This set is called the STANDARD BASIS for ℝ 3. In general, for ℝn the standard basis will have n vectors, each with a single 1 and zeroes elsewhere, so that they form the nxn identity matrix. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Linear Independence Example: Are the following sets of vectors linearly independent? Describe the span of each set. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Linear Independence Example: Are the following sets of vectors linearly independent? Describe the span of each set. V 4 spans a 2 -dimensional subset of ℝ 3 (a plane). Let’s reduce the augmented matrix for this one. The RREF matrix has a non-pivot column, which means there is a free variable. There are 2 pivot positions, so the span of this set must be 2 -dimensional (a plane in ℝ 3). This set is linearly dependent. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Linear Independence Example: Are the following sets of vectors linearly independent? Describe the span of each set. V 4 spans a 2 -dimensional subset of ℝ 3 (a plane). Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB