1. ## basis and dimension

Bit stuck step by step much appreciated

decide if each is a basis for r^3

(1) , (3)
(2) , (2)
(3) , (1)

and

(0), (1), (0)
(2), (1), (0)
(-1), (1), (0)

help much appreciated

Bit stuck step by step much appreciated

decide if each is a basis for r^3

(1) , (3)
(2) , (2)
(3) , (1)

and

(0), (1), (0)
(2), (1), (0)
(-1), (1), (0)

help much appreciated
The first one cannot be a basis for it has 2 elements. A basis for $\mathbb{R}^3$ will need 3 elements.

And the second one cannot be a basis for it contains the zero vector.
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 1, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9345922470092773, "perplexity": 4696.2323527660565}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187821189.10/warc/CC-MAIN-20171017125144-20171017144828-00031.warc.gz"}