B. The goal of this problem is to and he is of v. vand set of...

 

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B. The goal of this problem is to and he is of v. vand set of all linear V3; that is, find Span{V₁, V2, V3). Let y ER³ be an arbitrary vector. We want to know when there exist scalars C1, C2, C3 ER, such that Consider the three vectors v₁ V₂ V3 = G₁V₁ + ₂√₂ +03Vg = -8 in R³. Saying this another way, we want to find all the vectors that can be expressed as linear combinations of V₁, V2, and v₁. This may be all of R3 but it may be a plane in R³ instead. (a) The vector equation above gives a linear system of equations in the variables C₁, C₂, and ca. Write the linear system and the corresponding augmented matrix. (b) Find the echelon form of the augmented matrix from part (a). (Hint: You may want to swap rows 1 and 3 as your first row operation, then multiply row 1 by -1 as your second row operation. This should make the rest of your calculations easier to carry out). (c) Find the conditions on z. y, and for which this system has a solution. (d) Using the result from part (e), can you determine if V₁, V2, and va span all of R³? If they don't span R³, use the results of part (c) to give the equation of the plane that they span. B. The goal of this problem is to and he is of v. vand set of all linear V3; that is, find Span{V₁, V2, V3). Let y ER³ be an arbitrary vector. We want to know when there exist scalars C1, C2, C3 ER, such that Consider the three vectors v₁ V₂ V3 = G₁V₁ + ₂√₂ +03Vg = -8 in R³. Saying this another way, we want to find all the vectors that can be expressed as linear combinations of V₁, V2, and v₁. This may be all of R3 but it may be a plane in R³ instead. (a) The vector equation above gives a linear system of equations in the variables C₁, C₂, and ca. Write the linear system and the corresponding augmented matrix. (b) Find the echelon form of the augmented matrix from part (a). (Hint: You may want to swap rows 1 and 3 as your first row operation, then multiply row 1 by -1 as your second row operation. This should make the rest of your calculations easier to carry out). (c) Find the conditions on z. y, and for which this system has a solution. (d) Using the result from part (e), can you determine if V₁, V2, and va span all of R³? If they don't span R³, use the results of part (c) to give the equation of the plane that they span.